Theorem ax16i 2046

Description: Inference with ax16 2045 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)

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

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

Proof of Theorem ax16i

Step	Hyp	Ref	Expression
1		nfv 1619	. . 3 |- F/z x = y
2		nfv 1619	. . 3 |- F/x z = y
3		ax-8 1675	. . 3 |- (x = z -> (x = y -> z = y))
4	1, 2, 3	cbv3 1982	. 2 |- (A.x x = y -> A.z z = y)
5		ax-8 1675	. . . . 5 |- (z = x -> (z = y -> x = y))
6	5	spimv 1990	. . . 4 |- (A.z z = y -> x = y)
7		equcomi 1679	. . . . . 6 |- (x = y -> y = x)
8		equcomi 1679	. . . . . . 7 |- (z = y -> y = z)
9		ax-8 1675	. . . . . . 7 |- (y = z -> (y = x -> z = x))
10	8, 9	syl 15	. . . . . 6 |- (z = y -> (y = x -> z = x))
11	7, 10	syl5com 26	. . . . 5 |- (x = y -> (z = y -> z = x))
12	11	alimdv 1621	. . . 4 |- (x = y -> (A.z z = y -> A.z z = x))
13	6, 12	mpcom 32	. . 3 |- (A.z z = y -> A.z z = x)
14		equcomi 1679	. . . 4 |- (z = x -> x = z)
15	14	alimi 1559	. . 3 |- (A.z z = x -> A.z x = z)
16	13, 15	syl 15	. 2 |- (A.z z = y -> A.z x = z)
17		ax16i.1	. . . . 5 |- (x = z -> (ph <-> ps))
18	17	biimpcd 215	. . . 4 |- (ph -> (x = z -> ps))
19	18	alimdv 1621	. . 3 |- (ph -> (A.z x = z -> A.zps))
20		ax16i.2	. . . . 5 |- (ps -> A.xps)
21	20	nfi 1551	. . . 4 |- F/xps
22		nfv 1619	. . . 4 |- F/zph
23	17	biimprd 214	. . . . 5 |- (x = z -> (ps -> ph))
24	14, 23	syl 15	. . . 4 |- (z = x -> (ps -> ph))
25	21, 22, 24	cbv3 1982	. . 3 |- (A.zps -> A.xph)
26	19, 25	syl6com 31	. 2 |- (A.z x = z -> (ph -> A.xph))
27	4, 16, 26	3syl 18	1 |- (A.x x = y -> (ph -> A.xph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540

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

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:  ax16ALT  2047