Theorem equidq 1315

Description: equid 1484 with universal quantifier without using ax-4 1319 or ax-17 1317.

Assertion

Ref	Expression
equidq	|- A.y x = x

Proof of Theorem equidq

Step	Hyp	Ref	Expression
1		equidqe 1314	. 2 |- -. A.y -. x = x
2		ax-6 1303	. . 3 |- (-. A.y x = x -> A.y -. A.y x = x)
3		hbequid 1313	. . . . . 6 |- (x = x -> A.y x = x)
4	3	con3i 114	. . . . 5 |- (-. A.y x = x -> -. x = x)
5	4	ax-gen 1305	. . . 4 |- A.y(-. A.y x = x -> -. x = x)
6		ax-5 1302	. . . 4 |- (A.y(-. A.y x = x -> -. x = x) -> (A.y -. A.y x = x -> A.y -. x = x))
7	5, 6	ax-mp 7	. . 3 |- (A.y -. A.y x = x -> A.y -. x = x)
8	2, 7	syl 12	. 2 |- (-. A.y x = x -> A.y -. x = x)
9	1, 8	mt3 127	1 |- A.y x = x

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-5 1302  ax-6 1303  ax-gen 1305  ax-8 1306  ax-9 1307  ax-12 1310

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