Theorem ax11 1261

Description: Rederivation of axiom ax-11 1008 from the orginal version, ax-11o 1260. See theorem ax11o 1259 for the derivation of ax-11o 1260 from ax-11 1008.

This theorem should not be referenced in any proof. Instead, use ax-11 1008 above so that uses of ax-11 1008 can be more easily identified.

Assertion

Ref	Expression
ax11	|- (x = y -> (A.yph -> A.x(x = y -> ph)))

Proof of Theorem ax11

Step	Hyp	Ref	Expression
1		pm4.2d 178	. . . . 5 |- (A.x x = y -> (ph <-> ph))
2	1	dral1 1196	. . . 4 |- (A.x x = y -> (A.xph <-> A.yph))
3		ax-1 4	. . . . 5 |- (ph -> (x = y -> ph))
4	3	19.20i 1033	. . . 4 |- (A.xph -> A.x(x = y -> ph))
5	2, 4	syl6bir 222	. . 3 |- (A.x x = y -> (A.yph -> A.x(x = y -> ph)))
6	5	a1d 12	. 2 |- (A.x x = y -> (x = y -> (A.yph -> A.x(x = y -> ph))))
7		ax-11o 1260	. . 3 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
8		ax-4 1014	. . 3 |- (A.yph -> ph)
9	7, 8	syl7 23	. 2 |- (-. A.x x = y -> (x = y -> (A.yph -> A.x(x = y -> ph))))
10	6, 9	pm2.61i 132	1 |- (x = y -> (A.yph -> A.x(x = y -> ph)))

Syntax hints:  -. wn 2   -> wi 3  A.wal 995   = wceq 997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1003  ax-gen 1004  ax-10 1007  ax-12 1009  ax-4 1014  ax-5o 1016  ax-10o 1182  ax-11o 1260

This theorem depends on definitions:  df-bi 154  df-an 232

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