Theorem ax11inda2ALT 1369

Description: A proof of ax11inda2 1370 that is slightly more direct.

Hypothesis

Ref	Expression
ax11inda2.1	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Assertion

Ref	Expression
ax11inda2ALT	|- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

Distinct variable group:   y,z

Proof of Theorem ax11inda2ALT

Step	Hyp	Ref	Expression
1		ax-1 4	. . . . . . . 8 |- (A.xph -> (x = y -> A.xph))
2	1	a5i 989	. . . . . . 7 |- (A.xph -> A.x(x = y -> A.xph))
3	2	a1i 8	. . . . . 6 |- (A.z z = x -> (A.xph -> A.x(x = y -> A.xph)))
4		pm4.2d 171	. . . . . . 7 |- (A.z z = x -> (ph <-> ph))
5	4	dral1 1154	. . . . . 6 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 612	. . . . . . 7 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2 1155	. . . . . 6 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 543	. . . . 5 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	alequcoms 1143	. . . 4 |- (A.x x = z -> (A.zph -> A.x(x = y -> A.zph)))
10	9	a1d 12	. . 3 |- (A.x x = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
11	10	a1d 12	. 2 |- (A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
12		hbnae 1147	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
13		hba1 1003	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
14	12, 13	hban 1009	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> A.z(-. A.x x = y /\ A.z x = y))
15		ax11inda2.1	. . . . . . . 8 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
16	15	imp 350	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
17		ax-4 973	. . . . . . 7 |- (A.z x = y -> x = y)
18	16, 17	sylan2 451	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
19	14, 18	19.20d 996	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
20		simplr 413	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
21		dveeq1 1354	. . . . . . . 8 |- (-. A.z z = x -> (x = y -> A.z x = y))
22	21	nalequcoms 1144	. . . . . . 7 |- (-. A.x x = z -> (x = y -> A.z x = y))
23	22	imp 350	. . . . . 6 |- ((-. A.x x = z /\ x = y) -> A.z x = y)
24	23	adantlr 393	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> A.z x = y)
25	19, 20, 24	sylanc 471	. . . 4 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
26		hbnae 1147	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
27		hbnae 1147	. . . . . . . . 9 |- (-. A.x x = z -> A.z -. A.x x = z)
28	27, 22	19.21ai 998	. . . . . . . 8 |- (-. A.x x = z -> A.z(x = y -> A.z x = y))
29		19.21t 1115	. . . . . . . 8 |- (A.z(x = y -> A.z x = y) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
30	28, 29	syl 10	. . . . . . 7 |- (-. A.x x = z -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
31	26, 30	albid 1104	. . . . . 6 |- (-. A.x x = z -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
32		ax-7 962	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
33	31, 32	syl5bi 208	. . . . 5 |- (-. A.x x = z -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
34	33	ad2antrr 404	. . . 4 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
35	25, 34	syld 27	. . 3 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.x(x = y -> A.zph)))
36	35	exp31 376	. 2 |- (-. A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
37	11, 36	pm2.61i 126	1 |- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956

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-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140

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

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