Metamath Proof Explorer


Theorem axc11nfromc11

Description: Rederivation of ax-c11n from original version ax-c11 . See Theorem axc11 for the derivation of ax-c11 from ax-c11n .

This theorem should not be referenced in any proof. Instead, use ax-c11n above so that uses of ax-c11n can be more easily identified, or use aecom-o when this form is needed for studies involving ax-c11 and omitting ax-5 . (Contributed by NM, 16-May-2008) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion axc11nfromc11
|- ( A. x x = y -> A. y y = x )

Proof

Step Hyp Ref Expression
1 ax-c11
 |-  ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
2 1 pm2.43i
 |-  ( A. x x = y -> A. y x = y )
3 equcomi
 |-  ( x = y -> y = x )
4 3 alimi
 |-  ( A. y x = y -> A. y y = x )
5 2 4 syl
 |-  ( A. x x = y -> A. y y = x )