Theorem ax11 1589

Description: Rederivation of axiom ax-11 1309 from the orginal version, ax-11o 1588. See theorem ax11o 1587 for the derivation of ax-11o 1588 from ax-11 1309.

This theorem should not be referenced in any proof. Instead, use ax-11 1309 above so that uses of ax-11 1309 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		biidd 188	. . . . 5 |- (A.x x = y -> (ph <-> ph))
2	1	dral1 1515	. . . 4 |- (A.x x = y -> (A.xph <-> A.yph))
3		ax-1 4	. . . . 5 |- (ph -> (x = y -> ph))
4	3	alimi 1338	. . . 4 |- (A.xph -> A.x(x = y -> ph))
5	2, 4	syl6bir 232	. . 3 |- (A.x x = y -> (A.yph -> A.x(x = y -> ph)))
6	5	a1d 15	. 2 |- (A.x x = y -> (x = y -> (A.yph -> A.x(x = y -> ph))))
7		ax-11o 1588	. . 3 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
8		ax-4 1319	. . 3 |- (A.yph -> ph)
9	7, 8	syl7 26	. 2 |- (-. A.x x = y -> (x = y -> (A.yph -> A.x(x = y -> ph))))
10	6, 9	pm2.61i 140	1 |- (x = y -> (A.yph -> A.x(x = y -> ph)))

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-10 1308  ax-12 1310  ax-4 1319  ax-5o 1321  ax-10o 1500  ax-11o 1588

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

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