Theorem axc11n11 32672

Description: Proof of axc11n 2307 from { ax-1 6-- ax-7 1935, axc11 2314 } . Almost identical to axc11nfromc11 34211. (Contributed by NM, 6-Jul-2021.) (Proof modification is discouraged.)

Assertion

Ref	Expression
axc11n11	|- ( A. x x = y -> A. y y = x )

Proof of Theorem axc11n11

Step	Hyp	Ref	Expression
1		axc11 2314	. . 3 |- ( A. x x = y -> ( A. x x = y -> A. y x = y ) )
2	1	pm2.43i 52	. 2 |- ( A. x x = y -> A. y x = y )
3		equcomi 1944	. 2 |- ( x = y -> y = x )
4	2, 3	sylg 1750	1 |- ( A. x x = y -> A. y y = x )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)