Theorem ax11indalem 1410

Description: Lemma for ax11inda2 1412 and ax11inda 1413.

Hypothesis

Ref	Expression
ax11indalem.1	|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))

Assertion

Ref	Expression
ax11indalem	|- (-. A.y y = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))

Proof of Theorem ax11indalem

Step	Hyp	Ref	Expression
1		ax-1 4	. . . . . . . . 9 |- (A.xph -> (x = y -> A.xph))
2	1	a5i 1030	. . . . . . . 8 |- (A.xph -> A.x(x = y -> A.xph))
3	2	a1i 8	. . . . . . 7 |- (A.z z = x -> (A.xph -> A.x(x = y -> A.xph)))
4		pm4.2d 178	. . . . . . . 8 |- (A.z z = x -> (ph <-> ph))
5	4	dral1 1196	. . . . . . 7 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 623	. . . . . . . 8 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2 1197	. . . . . . 7 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 554	. . . . . 6 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	alequcoms 1185	. . . . 5 |- (A.x x = z -> (A.zph -> A.x(x = y -> A.zph)))
10	9	a1d 12	. . . 4 |- (A.x x = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
11	10	a1d 12	. . 3 |- (A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
12	11	adantr 398	. 2 |- ((A.x x = z /\ -. A.y y = z) -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
13		hbnae 1189	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
14		hba1 1044	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
15	13, 14	hban 1050	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> A.z(-. A.x x = y /\ A.z x = y))
16		ax11indalem.1	. . . . . . . 8 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
17	16	imp 357	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
18		ax-4 1014	. . . . . . 7 |- (A.z x = y -> x = y)
19	17, 18	sylan2 462	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
20	15, 19	19.20d 1037	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
21		simplr 422	. . . . 5 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
22		ax-12 1009	. . . . . . . . 9 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
23	22	imp 357	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
24		alequcom 1184	. . . . . . . . 9 |- (A.z z = x -> A.x x = z)
25	24	con3i 104	. . . . . . . 8 |- (-. A.x x = z -> -. A.z z = x)
26		alequcom 1184	. . . . . . . . 9 |- (A.z z = y -> A.y y = z)
27	26	con3i 104	. . . . . . . 8 |- (-. A.y y = z -> -. A.z z = y)
28	23, 25, 27	syl2an 465	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> (x = y -> A.z x = y))
29	28	imp 357	. . . . . 6 |- (((-. A.x x = z /\ -. A.y y = z) /\ x = y) -> A.z x = y)
30	29	adantlr 402	. . . . 5 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> A.z x = y)
31	20, 21, 30	sylanc 482	. . . 4 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
32		hbnae 1189	. . . . . . . 8 |- (-. A.x x = z -> A.x -. A.x x = z)
33		hbnae 1189	. . . . . . . 8 |- (-. A.y y = z -> A.x -. A.y y = z)
34	32, 33	hban 1050	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> A.x(-. A.x x = z /\ -. A.y y = z))
35		hbnae 1189	. . . . . . . . . 10 |- (-. A.x x = z -> A.z -. A.x x = z)
36		hbnae 1189	. . . . . . . . . 10 |- (-. A.y y = z -> A.z -. A.y y = z)
37	35, 36	hban 1050	. . . . . . . . 9 |- ((-. A.x x = z /\ -. A.y y = z) -> A.z(-. A.x x = z /\ -. A.y y = z))
38	37, 28	19.21ai 1039	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.y y = z) -> A.z(x = y -> A.z x = y))
39		19.21t 1156	. . . . . . . 8 |- (A.z(x = y -> A.z x = y) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
40	38, 39	syl 10	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
41	34, 40	albid 1145	. . . . . 6 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
42		ax-7 1003	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
43	41, 42	syl5bi 215	. . . . 5 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
44	43	ad2antrr 413	. . . 4 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
45	31, 44	syld 27	. . 3 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.x(x = y -> A.zph)))
46	45	exp31 385	. 2 |- ((-. A.x x = z /\ -. A.y y = z) -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
47	12, 46	pm2.61ian 487	1 |- (-. A.y y = z -> (-. 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 is referenced by:  ax11inda2 1412

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-10 1007  ax-12 1009  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-10o 1182

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

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