Theorem ax10 1710

Description: Rederivation of ax-10 1498 from original version ax-10o 1709. See Theorem ax10o 1708 for the derivation of ax-10o 1709 from ax-10 1498.

This theorem should not be referenced in any proof. Instead, use ax-10 1498 above so that uses of ax-10 1498 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 1709	. . 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 1697	. . 3 |- ( x = y -> y = x )
4	3	alimi 1448	. 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 1346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-5 1440  ax-gen 1442  ax-ie2 1487  ax-8 1497  ax-17 1519  ax-i9 1523  ax-10o 1709

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)