Theorem ax11inda2ALT 2198

Description: A proof of ax11inda2 2199 that is slightly more direct. (Contributed by NM, 4-May-2007.) (Proof modification is discouraged.) (New usage is discouraged.)

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

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem ax11inda2ALT

Step	Hyp	Ref	Expression
1		ax-1 6	. . . . . . . 8 |- (A.xph -> (x = y -> A.xph))
2	1	a5i-o 2150	. . . . . . 7 |- (A.xph -> A.x(x = y -> A.xph))
3	2	a1i 10	. . . . . 6 |- (A.z z = x -> (A.xph -> A.x(x = y -> A.xph)))
4		biidd 228	. . . . . . 7 |- (A.z z = x -> (ph <-> ph))
5	4	dral1-o 2154	. . . . . 6 |- (A.z z = x -> (A.zph <-> A.xph))
6	5	imbi2d 307	. . . . . . 7 |- (A.z z = x -> ((x = y -> A.zph) <-> (x = y -> A.xph)))
7	6	dral2-o 2181	. . . . . 6 |- (A.z z = x -> (A.x(x = y -> A.zph) <-> A.x(x = y -> A.xph)))
8	3, 5, 7	3imtr4d 259	. . . . 5 |- (A.z z = x -> (A.zph -> A.x(x = y -> A.zph)))
9	8	aecoms-o 2152	. . . 4 |- (A.x x = z -> (A.zph -> A.x(x = y -> A.zph)))
10	9	a1d 22	. . 3 |- (A.x x = z -> (x = y -> (A.zph -> A.x(x = y -> A.zph))))
11	10	a1d 22	. 2 |- (A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
12		simplr 731	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> -. A.x x = y)
13		dveeq1-o 2187	. . . . . . . 8 |- (-. A.z z = x -> (x = y -> A.z x = y))
14	13	naecoms-o 2178	. . . . . . 7 |- (-. A.x x = z -> (x = y -> A.z x = y))
15	14	imp 418	. . . . . 6 |- ((-. A.x x = z /\ x = y) -> A.z x = y)
16	15	adantlr 695	. . . . 5 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> A.z x = y)
17		hbnae-o 2179	. . . . . . 7 |- (-. A.x x = y -> A.z -. A.x x = y)
18		hba1-o 2149	. . . . . . 7 |- (A.z x = y -> A.zA.z x = y)
19	17, 18	hban 1828	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> A.z(-. A.x x = y /\ A.z x = y))
20		ax-4 2135	. . . . . . 7 |- (A.z x = y -> x = y)
21		ax11inda2.1	. . . . . . . 8 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
22	21	imp 418	. . . . . . 7 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
23	20, 22	sylan2 460	. . . . . 6 |- ((-. A.x x = y /\ A.z x = y) -> (ph -> A.x(x = y -> ph)))
24	19, 23	alimdh 1563	. . . . 5 |- ((-. A.x x = y /\ A.z x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
25	12, 16, 24	syl2anc 642	. . . 4 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.zA.x(x = y -> ph)))
26		ax-7 1734	. . . . . 6 |- (A.zA.x(x = y -> ph) -> A.xA.z(x = y -> ph))
27		hbnae-o 2179	. . . . . . 7 |- (-. A.x x = z -> A.x -. A.x x = z)
28		hbnae-o 2179	. . . . . . . . 9 |- (-. A.x x = z -> A.z -. A.x x = z)
29	28, 14	nfdh 1767	. . . . . . . 8 |- (-. A.x x = z -> F/z x = y)
30		19.21t 1795	. . . . . . . 8 |- ( F/z x = y -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
31	29, 30	syl 15	. . . . . . 7 |- (-. A.x x = z -> (A.z(x = y -> ph) <-> (x = y -> A.zph)))
32	27, 31	albidh 1590	. . . . . 6 |- (-. A.x x = z -> (A.xA.z(x = y -> ph) <-> A.x(x = y -> A.zph)))
33	26, 32	syl5ib 210	. . . . 5 |- (-. A.x x = z -> (A.zA.x(x = y -> ph) -> A.x(x = y -> A.zph)))
34	33	ad2antrr 706	. . . 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 40	. . 3 |- (((-. A.x x = z /\ -. A.x x = y) /\ x = y) -> (A.zph -> A.x(x = y -> A.zph)))
36	35	exp31 587	. 2 |- (-. A.x x = z -> (-. A.x x = y -> (x = y -> (A.zph -> A.x(x = y -> A.zph)))))
37	11, 36	pm2.61i 156	1 |- (-. 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: (None)