Theorem axrepndlem1 4963

Description: Lemma for the Axiom of Replacement with no distinct variable conditions.

Assertion

Ref	Expression
axrepndlem1	|- (-. A.y y = z -> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))

Distinct variable groups:   x,z   x,y

Proof of Theorem axrepndlem1

Step	Hyp	Ref	Expression
1		axrep2 2701	. 2 |- E.x(E.yA.w([w / z]ph -> w = y) -> A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph)))
2		hbnae 1149	. . 3 |- (-. A.y y = z -> A.x -. A.y y = z)
3		hbnae 1149	. . . . 5 |- (-. A.y y = z -> A.y -. A.y y = z)
4		hbnae 1149	. . . . . 6 |- (-. A.y y = z -> A.z -. A.y y = z)
5		ax-17 973	. . . . . . . . 9 |- (ph -> A.wph)
6	5	hbsb3 1208	. . . . . . . 8 |- ([w / z]ph -> A.z[w / z]ph)
7	6	a1i 8	. . . . . . 7 |- (-. A.y y = z -> ([w / z]ph -> A.z[w / z]ph))
8		nd5 4961	. . . . . . 7 |- (-. A.y y = z -> (w = y -> A.z w = y))
9	4, 7, 8	hbimd 1112	. . . . . 6 |- (-. A.y y = z -> (([w / z]ph -> w = y) -> A.z([w / z]ph -> w = y)))
10		sbequ12r 1184	. . . . . . . 8 |- (w = z -> ([w / z]ph <-> ph))
11		equequ1 1136	. . . . . . . 8 |- (w = z -> (w = y <-> z = y))
12	10, 11	imbi12d 628	. . . . . . 7 |- (w = z -> (([w / z]ph -> w = y) <-> (ph -> z = y)))
13	12	a1i 8	. . . . . 6 |- (-. A.y y = z -> (w = z -> (([w / z]ph -> w = y) <-> (ph -> z = y))))
14	4, 9, 13	cbvald 1322	. . . . 5 |- (-. A.y y = z -> (A.w([w / z]ph -> w = y) <-> A.z(ph -> z = y)))
15	3, 14	exbid 1107	. . . 4 |- (-. A.y y = z -> (E.yA.w([w / z]ph -> w = y) <-> E.yA.z(ph -> z = y)))
16		ax-17 973	. . . . . . 7 |- (w e. x -> A.z w e. x)
17	16	a1i 8	. . . . . 6 |- (-. A.y y = z -> (w e. x -> A.z w e. x))
18		dveel2 1359	. . . . . . . . 9 |- (-. A.z z = y -> (x e. y -> A.z x e. y))
19	18	nalequcoms 1146	. . . . . . . 8 |- (-. A.y y = z -> (x e. y -> A.z x e. y))
20	6	hbal 1007	. . . . . . . . 9 |- (A.y[w / z]ph -> A.zA.y[w / z]ph)
21	20	a1i 8	. . . . . . . 8 |- (-. A.y y = z -> (A.y[w / z]ph -> A.zA.y[w / z]ph))
22	19, 21	hband 1113	. . . . . . 7 |- (-. A.y y = z -> ((x e. y /\ A.y[w / z]ph) -> A.z(x e. y /\ A.y[w / z]ph)))
23	2, 22	hbexd 1116	. . . . . 6 |- (-. A.y y = z -> (E.x(x e. y /\ A.y[w / z]ph) -> A.zE.x(x e. y /\ A.y[w / z]ph)))
24	4, 17, 23	hbbid 1114	. . . . 5 |- (-. A.y y = z -> ((w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) -> A.z(w e. x <-> E.x(x e. y /\ A.y[w / z]ph))))
25		elequ1 1138	. . . . . . . 8 |- (w = z -> (w e. x <-> z e. x))
26	25	adantl 390	. . . . . . 7 |- ((-. A.y y = z /\ w = z) -> (w e. x <-> z e. x))
27		dveeq2 1214	. . . . . . . . . . 11 |- (-. A.y y = z -> (w = z -> A.y w = z))
28	27	imp 350	. . . . . . . . . 10 |- ((-. A.y y = z /\ w = z) -> A.y w = z)
29		hba1 1005	. . . . . . . . . . 11 |- (A.y w = z -> A.yA.y w = z)
30	10	a4s 986	. . . . . . . . . . 11 |- (A.y w = z -> ([w / z]ph <-> ph))
31	29, 30	albid 1106	. . . . . . . . . 10 |- (A.y w = z -> (A.y[w / z]ph <-> A.yph))
32	28, 31	syl 10	. . . . . . . . 9 |- ((-. A.y y = z /\ w = z) -> (A.y[w / z]ph <-> A.yph))
33	32	anbi2d 618	. . . . . . . 8 |- ((-. A.y y = z /\ w = z) -> ((x e. y /\ A.y[w / z]ph) <-> (x e. y /\ A.yph)))
34	33	exbidv 1281	. . . . . . 7 |- ((-. A.y y = z /\ w = z) -> (E.x(x e. y /\ A.y[w / z]ph) <-> E.x(x e. y /\ A.yph)))
35	26, 34	bibi12d 631	. . . . . 6 |- ((-. A.y y = z /\ w = z) -> ((w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) <-> (z e. x <-> E.x(x e. y /\ A.yph))))
36	35	ex 373	. . . . 5 |- (-. A.y y = z -> (w = z -> ((w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) <-> (z e. x <-> E.x(x e. y /\ A.yph)))))
37	4, 24, 36	cbvald 1322	. . . 4 |- (-. A.y y = z -> (A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph)) <-> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
38	15, 37	imbi12d 628	. . 3 |- (-. A.y y = z -> ((E.yA.w([w / z]ph -> w = y) -> A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph))) <-> (E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))))
39	2, 38	exbid 1107	. 2 |- (-. A.y y = z -> (E.x(E.yA.w([w / z]ph -> w = y) -> A.w(w e. x <-> E.x(x e. y /\ A.y[w / z]ph))) <-> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))))
40	1, 39	mpbii 193	1 |- (-. A.y y = z -> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  E.wex 982  [wsbc 1172

This theorem is referenced by:  axrepndlem2 4964

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-rep 2699

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174

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