Theorem axc11nfromc11 32409

Description: Rederivation of ax-c11n 32372 from original version ax-c11 32371. See theorem axc11 2120 for the derivation of ax-c11 32371 from ax-c11n 32372.

This theorem should not be referenced in any proof. Instead, use ax-c11n 32372 above so that uses of ax-c11n 32372 can be more easily identified, or use aecom-o 32383 when this form is needed for studies involving ax-c11 32371 and omitting ax-5 1752. (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 32371	. . 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 1847	. . 3 |- ( x = y -> y = x )
4	3	alimi 1678	. 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 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-c11 32371

This theorem depends on definitions:  df-bi 188  df-ex 1658

This theorem is referenced by: (None)