Theorem ax11inda2ALT 1411

Description: A proof of ax11inda2 1412 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 1030	. . . . . . 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 178	. . . . . . 7 |- (A.z z = x -> (ph <-> ph))
5	4	dral1 1196	. . . . . 6 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 623	. . . . . . 7 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2 1197	. . . . . 6 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 554	. . . . 5 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	alequcoms 1185	. . . 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 1189	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
13		hba1 1044	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
14	12, 13	hban 1050	. . . . . 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 357	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
17		ax-4 1014	. . . . . . 7 |- (A.z x = y -> x = y)
18	16, 17	sylan2 462	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
19	14, 18	19.20d 1037	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
20		simplr 422	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
21		dveeq1 1396	. . . . . . . 8 |- (-. A.z z = x -> (x = y -> A.z x = y))
22	21	nalequcoms 1186	. . . . . . 7 |- (-. A.x x = z -> (x = y -> A.z x = y))
23	22	imp 357	. . . . . 6 |- ((-. A.x x = z /\ x = y) -> A.z x = y)
24	23	adantlr 402	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> A.z x = y)
25	19, 20, 24	sylanc 482	. . . 4 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
26		hbnae 1189	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
27		hbnae 1189	. . . . . . . . 9 |- (-. A.x x = z -> A.z -. A.x x = z)
28	27, 22	19.21ai 1039	. . . . . . . 8 |- (-. A.x x = z -> A.z(x = y -> A.z x = y))
29		19.21t 1156	. . . . . . . 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 1145	. . . . . 6 |- (-. A.x x = z -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
32		ax-7 1003	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
33	31, 32	syl5bi 215	. . . . 5 |- (-. A.x x = z -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
34	33	ad2antrr 413	. . . 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 385	. 2 |- (-. A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
37	11, 36	pm2.61i 132	1 |- (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 153   /\ wa 230  A.wal 995   = wceq 997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-8 1005  ax-10 1007  ax-12 1009  ax-17 1012  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182

This theorem depends on definitions:  df-bi 154  df-an 232

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