Theorem ax10 1695

Description: Rederivation of ax-10 1483 from original version ax-10o 1694. See theorem ax10o 1693 for the derivation of ax-10o 1694 from ax-10 1483.

This theorem should not be referenced in any proof. Instead, use ax-10 1483 above so that uses of ax-10 1483 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 1694	. . 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 1680	. . 3 |- ( x = y -> y = x )
4	3	alimi 1431	. 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 1329

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1423  ax-gen 1425  ax-ie2 1470  ax-8 1482  ax-17 1506  ax-i9 1510  ax-10o 1694

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)