Theorem a16g 1278

Description: A generalization of axiom ax-16 1212.

Assertion

Ref	Expression
a16g	|- (A.x x = y -> (ph -> A.zph))

Distinct variable group:   x,y

Proof of Theorem a16g

Step	Hyp	Ref	Expression
1		hbae 1147	. . 3 |- (A.x x = y -> A.zA.x x = y)
2		ax-9 967	. . . . 5 |- -. A.x -. x = z
3		ax-16 1212	. . . . 5 |- (A.x x = y -> (-. x = z -> A.x -. x = z))
4	2, 3	mt3i 113	. . . 4 |- (A.x x = y -> x = z)
5		equcomi 1130	. . . 4 |- (x = z -> z = x)
6	4, 5	syl 10	. . 3 |- (A.x x = y -> z = x)
7	1, 6	19.21ai 1000	. 2 |- (A.x x = y -> A.z z = x)
8		ax-16 1212	. . 3 |- (A.x x = y -> (ph -> A.xph))
9		pm4.2d 171	. . . . 5 |- (A.z z = x -> (ph <-> ph))
10	9	dral1 1156	. . . 4 |- (A.z z = x -> (A.zph <-> A.xph))
11	10	biimprd 154	. . 3 |- (A.z z = x -> (A.xph -> A.zph))
12	8, 11	syl9r 58	. 2 |- (A.z z = x -> (A.x x = y -> (ph -> A.zph)))
13	7, 12	mpcom 49	1 |- (A.x x = y -> (ph -> A.zph))

Syntax hints:  -. wn 2   -> wi 3  A.wal 956   = wceq 958

This theorem is referenced by:  a16gb 1279  ax11inda2 1372

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-10 968  ax-12 970  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212

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

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