Theorem axc11nfromc11 32541

Description: Rederivation of ax-c11n 32504 from original version ax-c11 32503. See theorem axc11 2158 for the derivation of ax-c11 32503 from ax-c11n 32504.

This theorem should not be referenced in any proof. Instead, use ax-c11n 32504 above so that uses of ax-c11n 32504 can be more easily identified, or use aecom-o 32515 when this form is needed for studies involving ax-c11 32503 and omitting ax-5 1768. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
axc11nfromc11	|- ( A. x x = y -> A. y y = x )

Proof of Theorem axc11nfromc11

Step	Hyp	Ref	Expression
1		ax-c11 32503	. . 3 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
2	1	pm2.43i 49	. 2 |- ( A. x x = y -> A. y x = y )
3		equcomi 1871	. . 3 |- ( x = y -> y = x )
4	3	alimi 1694	. 2 |- ( A. y x = y -> A. y y = x )
5	2, 4	syl 17	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -> wi 4   A.wal 1452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-c11 32503

This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674

This theorem is referenced by: (None)