Theorem axc11nfromc11 2258

Description: Rederivation of ax-c11n 2221 from original version ax-c11 2220. See theorem axc11 2058 for the derivation of ax-c11 2220 from ax-c11n 2221.

This theorem should not be referenced in any proof. Instead, use ax-c11n 2221 above so that uses of ax-c11n 2221 can be more easily identified, or use aecom-o 2232 when this form is needed for studies involving ax-c11 2220 and omitting ax-5 1709. (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 2220	. . 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 1798	. . 3 |- ( x = y -> y = x )
4	3	alimi 1638	. 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 1396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-c11 2220

This theorem depends on definitions:  df-bi 185  df-ex 1618

This theorem is referenced by: (None)