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Theorem bj-naecomsv 33399
Description: Version of naecoms 2026 with a dv condition, which does not require ax-13 1968. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
bj-naecomsv.1  |-  ( -. 
A. x  x  =  y  ->  ph )
Assertion
Ref Expression
bj-naecomsv  |-  ( -. 
A. y  y  =  x  ->  ph )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bj-naecomsv
StepHypRef Expression
1 bj-axc11nv 33397 . . 3  |-  ( A. x  x  =  y  ->  A. y  y  =  x )
2 bj-naecomsv.1 . . 3  |-  ( -. 
A. x  x  =  y  ->  ph )
31, 2nsyl4 142 . 2  |-  ( -. 
ph  ->  A. y  y  =  x )
43con1i 129 1  |-  ( -. 
A. y  y  =  x  ->  ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-12 1803
This theorem depends on definitions:  df-bi 185  df-ex 1597
This theorem is referenced by: (None)
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