Theorem ax11indalem 2197

Description: Lemma for ax11inda2 2199 and ax11inda 2200. (Contributed by NM, 24-Jan-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

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 6	. . . . . . . . 9 |- (A.xph -> (x = y -> A.xph))
2	1	a5i-o 2150	. . . . . . . 8 |- (A.xph -> A.x(x = y -> A.xph))
3	2	a1i 10	. . . . . . 7 |- (A.z z = x -> (A.xph -> A.x(x = y -> A.xph)))
4		biidd 228	. . . . . . . 8 |- (A.z z = x -> (ph <-> ph))
5	4	dral1-o 2154	. . . . . . 7 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 307	. . . . . . . 8 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2-o 2181	. . . . . . 7 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 259	. . . . . 6 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	aecoms-o 2152	. . . . 5 |- (A.x x = z -> (A.zph -> A.x(x = y -> A.zph)))
10	9	a1d 22	. . . 4 |- (A.x x = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
11	10	a1d 22	. . 3 |- (A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
12	11	adantr 451	. 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 731	. . . . 5 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
14		aecom-o 2151	. . . . . . . . 9 |- (A.z z = x -> A.x x = z)
15	14	con3i 127	. . . . . . . 8 |- (-. A.x x = z -> -. A.z z = x)
16		aecom-o 2151	. . . . . . . . 9 |- (A.z z = y -> A.y y = z)
17	16	con3i 127	. . . . . . . 8 |- (-. A.y y = z -> -. A.z z = y)
18		ax12o 1934	. . . . . . . . 9 |- (-. A.z z = x -> (-. A.z z = y -> (x = y -> A.z x = y)))
19	18	imp 418	. . . . . . . 8 |- ((-. A.z z = x /\ -. A.z z = y) -> (x = y -> A.z x = y))
20	15, 17, 19	syl2an 463	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> (x = y -> A.z x = y))
21	20	imp 418	. . . . . 6 |- (((-. A.x x = z /\ -. A.y y = z) /\ x = y) -> A.z x = y)
22	21	adantlr 695	. . . . 5 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> A.z x = y)
23		hbnae-o 2179	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
24		hba1-o 2149	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
25	23, 24	hban 1828	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> A.z(-. A.x x = y /\ A.z x = y))
26		ax-4 2135	. . . . . . 7 |- (A.z x = y -> x = y)
27		ax11indalem.1	. . . . . . . 8 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
28	27	imp 418	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
29	26, 28	sylan2 460	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
30	25, 29	alimdh 1563	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
31	13, 22, 30	syl2anc 642	. . . 4 |- ((((-. A.x x = z /\ -. A.y y = z) /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
32		ax-7 1734	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
33		hbnae-o 2179	. . . . . . . 8 |- (-. A.x x = z -> A.x -. A.x x = z)
34		hbnae-o 2179	. . . . . . . 8 |- (-. A.y y = z -> A.x -. A.y y = z)
35	33, 34	hban 1828	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> A.x(-. A.x x = z /\ -. A.y y = z))
36		hbnae-o 2179	. . . . . . . . . 10 |- (-. A.x x = z -> A.z -. A.x x = z)
37		hbnae-o 2179	. . . . . . . . . 10 |- (-. A.y y = z -> A.z -. A.y y = z)
38	36, 37	hban 1828	. . . . . . . . 9 |- ((-. A.x x = z /\ -. A.y y = z) -> A.z(-. A.x x = z /\ -. A.y y = z))
39	38, 20	nfdh 1767	. . . . . . . 8 |- ((-. A.x x = z /\ -. A.y y = z) -> F/z x = y)
40		19.21t 1795	. . . . . . . 8 |- ( F/z x = y -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
41	39, 40	syl 15	. . . . . . 7 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
42	35, 41	albidh 1590	. . . . . 6 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
43	32, 42	syl5ib 210	. . . . 5 |- ((-. A.x x = z /\ -. A.y y = z) -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
44	43	ad2antrr 706	. . . 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 40	. . 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 587	. 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 765	1 |- (-. A.y y = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-4 2135  ax-5o 2136  ax-6o 2137  ax-10o 2139  ax-12o 2142

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ax11inda2  2199