Theorem axrep2 2693

Description: Axiom of Replacement expressed with the fewest number of different variables and without any restrictions on ph.

Assertion

Ref	Expression
axrep2	|- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))

Distinct variable group:   x,y,z

Proof of Theorem axrep2

Step	Hyp	Ref	Expression
1		hbe1 1016	. . . . 5 |- (E.wA.z(A.yph -> z = w) -> A.wE.wA.z(A.yph -> z = w))
2		ax-17 971	. . . . 5 |- (A.z(z e. x <-> E.x(x e. y /\ A.yph)) -> A.wA.z(z e. x <-> E.x(x e. y /\ A.yph)))
3	1, 2	hbim 1007	. . . 4 |- ((E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))) -> A.w(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
4	3	hbex 1006	. . 3 |- (E.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))) -> A.wE.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
5		elequ2 1137	. . . . . . . . 9 |- (w = y -> (x e. w <-> x e. y))
6	5	anbi1d 617	. . . . . . . 8 |- (w = y -> ((x e. w /\ A.yph) <-> (x e. y /\ A.yph)))
7	6	exbidv 1279	. . . . . . 7 |- (w = y -> (E.x(x e. w /\ A.yph) <-> E.x(x e. y /\ A.yph)))
8	7	bibi2d 618	. . . . . 6 |- (w = y -> ((z e. x <-> E.x(x e. w /\ A.yph)) <-> (z e. x <-> E.x(x e. y /\ A.yph))))
9	8	albidv 1278	. . . . 5 |- (w = y -> (A.z(z e. x <-> E.x(x e. w /\ A.yph)) <-> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
10	9	imbi2d 612	. . . 4 |- (w = y -> ((E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. w /\ A.yph))) <-> (E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))))
11	10	exbidv 1279	. . 3 |- (w = y -> (E.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. w /\ A.yph))) <-> E.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))))
12		axrep1 2692	. . 3 |- E.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. w /\ A.yph)))
13	4, 11, 12	chvar 1167	. 2 |- E.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))
14		ax-4 973	. . . . . . . 8 |- (A.yph -> ph)
15	14	imim1i 16	. . . . . . 7 |- ((ph -> z = y) -> (A.yph -> z = y))
16	15	19.20i 992	. . . . . 6 |- (A.z(ph -> z = y) -> A.z(A.yph -> z = y))
17	16	19.22i 1040	. . . . 5 |- (E.yA.z(ph -> z = y) -> E.yA.z(A.yph -> z = y))
18		ax-17 971	. . . . . 6 |- (A.z(A.yph -> z = y) -> A.wA.z(A.yph -> z = y))
19		hba1 1003	. . . . . . . 8 |- (A.yph -> A.yA.yph)
20		ax-17 971	. . . . . . . 8 |- (z = w -> A.y z = w)
21	19, 20	hbim 1007	. . . . . . 7 |- ((A.yph -> z = w) -> A.y(A.yph -> z = w))
22	21	hbal 1005	. . . . . 6 |- (A.z(A.yph -> z = w) -> A.yA.z(A.yph -> z = w))
23		equequ2 1135	. . . . . . . 8 |- (y = w -> (z = y <-> z = w))
24	23	imbi2d 612	. . . . . . 7 |- (y = w -> ((A.yph -> z = y) <-> (A.yph -> z = w)))
25	24	albidv 1278	. . . . . 6 |- (y = w -> (A.z(A.yph -> z = y) <-> A.z(A.yph -> z = w)))
26	18, 22, 25	cbvex 1166	. . . . 5 |- (E.yA.z(A.yph -> z = y) <-> E.wA.z(A.yph -> z = w))
27	17, 26	sylib 198	. . . 4 |- (E.yA.z(ph -> z = y) -> E.wA.z(A.yph -> z = w))
28	27	imim1i 16	. . 3 |- ((E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))) -> (E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
29	28	19.22i 1040	. 2 |- (E.x(E.wA.z(A.yph -> z = w) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))) -> E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph))))
30	13, 29	ax-mp 7	1 |- E.x(E.yA.z(ph -> z = y) -> A.z(z e. x <-> E.x(x e. y /\ A.yph)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980

This theorem is referenced by:  axrep3 2694  axrepndlem1 4932

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-rep 2691

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

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