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Theorem ax7w7AUX7 29590
Description: Special case of ax-7 1749. (Contributed by NM, 12-Oct-2017.)
Assertion
Ref Expression
ax7w7AUX7  |-  ( A. x A. y  -.  x  =  y  ->  A. y A. x  -.  x  =  y )

Proof of Theorem ax7w7AUX7
StepHypRef Expression
1 sp 1763 . . 3  |-  ( A. y  -.  x  =  y  ->  -.  x  =  y )
21alimi 1568 . 2  |-  ( A. x A. y  -.  x  =  y  ->  A. x  -.  x  =  y
)
3 ax9NEW7 29405 . . 3  |-  -.  A. x  -.  x  =  y
43pm2.21i 125 . 2  |-  ( A. x  -.  x  =  y  ->  A. y A. x  -.  x  =  y
)
52, 4syl 16 1  |-  ( A. x A. y  -.  x  =  y  ->  A. y A. x  -.  x  =  y )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950  ax-7v 29379
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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