Theorem a16gOLD 1654

Description: A generalization of axiom ax-16 1580.

Assertion

Ref	Expression
a16gOLD	|- (A.x x = y -> (ph -> A.zph))

Distinct variable group:   x,y

Proof of Theorem a16gOLD

Step	Hyp	Ref	Expression
1		hbae 1505	. . 3 |- (A.x x = y -> A.zA.x x = y)
2		ax-9 1307	. . . . 5 |- -. A.x -. x = z
3		ax-16 1580	. . . . 5 |- (A.x x = y -> (-. x = z -> A.x -. x = z))
4	2, 3	mt3i 128	. . . 4 |- (A.x x = y -> x = z)
5		equcomi 1487	. . . 4 |- (x = z -> z = x)
6	4, 5	syl 12	. . 3 |- (A.x x = y -> z = x)
7	1, 6	19.21ai 1345	. 2 |- (A.x x = y -> A.z z = x)
8		ax-16 1580	. 2 |- (A.x x = y -> (ph -> A.xph))
9		biidd 188	. . . 4 |- (A.z z = x -> (ph <-> ph))
10	9	dral1 1515	. . 3 |- (A.z z = x -> (A.zph <-> A.xph))
11	10	biimprd 171	. 2 |- (A.z z = x -> (A.xph -> A.zph))
12	7, 8, 11	sylsyld 32	1 |- (A.x x = y -> (ph -> A.zph))

Syntax hints:  -. wn 2   -> wi 3  A.wal 1296   = wceq 1298

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-8 1306  ax-9 1307  ax-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580

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

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