Theorem axc11nfromc11 2249

Description: Rederivation of ax-c11n 2212 from original version ax-c11 2211. See theorem axc11 2027 for the derivation of ax-c11 2211 from ax-c11n 2212.

This theorem should not be referenced in any proof. Instead, use ax-c11n 2212 above so that uses of ax-c11n 2212 can be more easily identified, or use aecom-o 2223 when this form is needed for studies involving ax-c11 2211 and omitting ax-5 1680. (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 2211	. . 3 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
2	1	pm2.43i 47	. 2 |- ( A. x x = y -> A. y x = y )
3		equcomi 1742	. . 3 |- ( x = y -> y = x )
4	3	alimi 1614	. 2 |- ( A. y x = y -> A. y y = x )
5	2, 4	syl 16	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -> wi 4   A.wal 1377

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-c11 2211

This theorem depends on definitions:  df-bi 185  df-ex 1597

This theorem is referenced by: (None)