Theorem ax10 1183

Description: Rederivation of ax-10 1007 from original version ax-10o 1182. See theorem ax10o 1181 for the derivation of ax-10o 1182 from ax-10 1007.

This theorem should not be referenced in any proof. Instead, use ax-10 1007 above so that uses of ax-10 1007 can be more easily identified.

Assertion

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

Proof of Theorem ax10

Step	Hyp	Ref	Expression
1		ax-10o 1182	. . 3 |- (A.x x = y -> (A.x x = y -> A.y x = y))
2	1	pm2.43i 64	. 2 |- (A.x x = y -> A.y x = y)
3		equcomi 1170	. . 3 |- (x = y -> y = x)
4	3	19.20i 1033	. 2 |- (A.y x = y -> A.y y = x)
5	2, 4	syl 10	1 |- (A.x x = y -> A.y y = x)

Syntax hints:   -> wi 3  A.wal 995   = wceq 997

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1004  ax-8 1005  ax-12 1009  ax-4 1014  ax-5o 1016  ax-6o 1019  ax-9o 1164  ax-10o 1182

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