Theorem ax10 1645

Description: Rederivation of ax-10 1436 from original version ax-10o 1644. See theorem ax10o 1643 for the derivation of ax-10o 1644 from ax-10 1436.

This theorem should not be referenced in any proof. Instead, use ax-10 1436 above so that uses of ax-10 1436 can be more easily identified. (Contributed by NM, 16-May-2008.) (New usage is discouraged.)

Assertion

Ref	Expression
ax10	|- ( A. x x = y -> A. y y = x )

Proof of Theorem ax10

Step	Hyp	Ref	Expression
1		ax-10o 1644	. . 3 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
2	1	pm2.43i 48	. 2 |- ( A. x x = y -> A. y x = y )
3		equcomi 1632	. . 3 |- ( x = y -> y = x )
4	3	alimi 1384	. 2 |- ( A. y x = y -> A. y y = x )
5	2, 4	syl 14	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -> wi 4   A.wal 1282

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-5 1376  ax-gen 1378  ax-ie2 1423  ax-8 1435  ax-17 1459  ax-i9 1463  ax-10o 1644

This theorem depends on definitions:  df-bi 115

This theorem is referenced by: (None)