Theorem ax16i 1272

Description: Inference with ax-16 1212 as its conclusion, that doesn't require ax-10 968, ax-11 969, or ax-12 970 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases.

Hypotheses

Ref	Expression
ax16i.1	|- (x = z -> (ph <-> ps))
ax16i.2	|- (ps -> A.xps)

Assertion

Ref	Expression
ax16i	|- (A.x x = y -> (ph -> A.xph))

Distinct variable groups:   x,y,z   ph,z

Proof of Theorem ax16i

Step	Hyp	Ref	Expression
1		ax-17 973	. . . 4 |- (x = y -> A.z x = y)
2		ax-17 973	. . . 4 |- (z = y -> A.x z = y)
3		ax-8 966	. . . 4 |- (x = z -> (x = y -> z = y))
4	1, 2, 3	cbv3 1166	. . 3 |- (A.x x = y -> A.z z = y)
5		ax-8 966	. . . . . 6 |- (z = x -> (z = y -> x = y))
6	5	a4imv 1209	. . . . 5 |- (A.z z = y -> x = y)
7		equid1 1271	. . . . . . . . 9 |- z = z
8		ax-8 966	. . . . . . . . 9 |- (z = y -> (z = z -> y = z))
9	7, 8	mpi 44	. . . . . . . 8 |- (z = y -> y = z)
10		ax-8 966	. . . . . . . 8 |- (y = z -> (y = x -> z = x))
11	9, 10	syl 10	. . . . . . 7 |- (z = y -> (y = x -> z = x))
12		equid1 1271	. . . . . . . 8 |- x = x
13		ax-8 966	. . . . . . . 8 |- (x = y -> (x = x -> y = x))
14	12, 13	mpi 44	. . . . . . 7 |- (x = y -> y = x)
15	11, 14	syl5com 52	. . . . . 6 |- (x = y -> (z = y -> z = x))
16	1, 15	19.20d 998	. . . . 5 |- (x = y -> (A.z z = y -> A.z z = x))
17	6, 16	mpcom 49	. . . 4 |- (A.z z = y -> A.z z = x)
18		ax-8 966	. . . . . 6 |- (z = x -> (z = z -> x = z))
19	7, 18	mpi 44	. . . . 5 |- (z = x -> x = z)
20	19	19.20i 994	. . . 4 |- (A.z z = x -> A.z x = z)
21	17, 20	syl 10	. . 3 |- (A.z z = y -> A.z x = z)
22	4, 21	syl 10	. 2 |- (A.x x = y -> A.z x = z)
23		ax-17 973	. . . 4 |- (ph -> A.zph)
24		ax16i.1	. . . . 5 |- (x = z -> (ph <-> ps))
25	24	biimpcd 155	. . . 4 |- (ph -> (x = z -> ps))
26	23, 25	19.20d 998	. . 3 |- (ph -> (A.z x = z -> A.zps))
27		ax16i.2	. . . 4 |- (ps -> A.xps)
28	24	biimprd 154	. . . . 5 |- (x = z -> (ps -> ph))
29	19, 28	syl 10	. . . 4 |- (z = x -> (ps -> ph))
30	27, 23, 29	cbv3 1166	. . 3 |- (A.zps -> A.xph)
31	26, 30	syl6com 53	. 2 |- (A.z x = z -> (ph -> A.xph))
32	22, 31	syl 10	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 956   = wceq 958

This theorem is referenced by:  ax16ALT 1273

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983

Copyright terms: Public domain