Theorem ax11indalem 1759

Description: Lemma for ax11inda2 1761 and ax11inda 1762.

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 1335	. . . . . . . 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		biidd 188	. . . . . . . 8 |- (A.z z = x -> (ph <-> ph))
5	4	dral1 1515	. . . . . . 7 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 674	. . . . . . . 8 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2 1516	. . . . . . 7 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 602	. . . . . 6 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	alequcoms 1503	. . . . 5 |- (A.x x = z -> (A.zph -> A.x(x = y -> A.zph)))
10	9	a1d 15	. . . 4 |- (A.x x = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
11	10	a1d 15	. . 3 |- (A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
12	11	adantr 425	. 2 |- ((A.x x = z /\ -. A.y y = z) -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
13		simplr 449	. . . . 5 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
14		ax-12 1310	. . . . . . . . 9 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
15	14	imp 377	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
16		alequcom 1502	. . . . . . . . 9 |- (A.z z = x -> A.x x = z)
17	16	con3i 114	. . . . . . . 8 |- (-. A.x x = z -> -. A.z z = x)
18		alequcom 1502	. . . . . . . . 9 |- (A.z z = y -> A.y y = z)
19	18	con3i 114	. . . . . . . 8 |- (-. A.y y = z -> -. A.z z = y)
20	15, 17, 19	syl2an 503	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> (x = y -> A.z x = y))
21	20	imp 377	. . . . . 6 |- (((-. A.x x = z /\ -. A.y y = z) /\ x = y) -> A.z x = y)
22	21	adantlr 429	. . . . 5 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> A.z x = y)
23		hbnae 1507	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
24		hba1 1350	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
25	23, 24	hban 1356	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> A.z(-. A.x x = y /\ A.z x = y))
26		ax11indalem.1	. . . . . . . 8 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
27	26	imp 377	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
28		ax-4 1319	. . . . . . 7 |- (A.z x = y -> x = y)
29	27, 28	sylan2 500	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
30	25, 29	alimd 1343	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
31	13, 22, 30	syl11anc 524	. . . 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 1507	. . . . . . . 8 |- (-. A.x x = z -> A.x -. A.x x = z)
33		hbnae 1507	. . . . . . . 8 |- (-. A.y y = z -> A.x -. A.y y = z)
34	32, 33	hban 1356	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> A.x(-. A.x x = z /\ -. A.y y = z))
35		hbnae 1507	. . . . . . . . . 10 |- (-. A.x x = z -> A.z -. A.x x = z)
36		hbnae 1507	. . . . . . . . . 10 |- (-. A.y y = z -> A.z -. A.y y = z)
37	35, 36	hban 1356	. . . . . . . . 9 |- ((-. A.x x = z /\ -. A.y y = z) -> A.z(-. A.x x = z /\ -. A.y y = z))
38	37, 20	19.21ai 1345	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.y y = z) -> A.z(x = y -> A.z x = y))
39		19.21t 1473	. . . . . . . 8 |- (A.z(x = y -> A.z x = y) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
40	38, 39	syl 12	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
41	34, 40	albid 1459	. . . . . 6 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
42		ax-7 1304	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
43	41, 42	syl5bi 225	. . . . 5 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
44	43	ad2antrr 440	. . . 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 30	. . 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 407	. 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 534	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 163   /\ wa 240  A.wal 1296   = wceq 1298

This theorem is referenced by:  ax11inda2 1761

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-10o 1500

This theorem depends on definitions:  df-bi 164  df-an 242

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