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Theorem bj-equs4v 33409
Description: Version of equs4 2008 with a dv condition, which does not require ax-13 1968 (neither ax-5 1680 nor ax-7 1739 nor ax-12 1803). (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-equs4v  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem bj-equs4v
StepHypRef Expression
1 ax6ev 1721 . 2  |-  E. x  x  =  y
2 exintr 1678 . 2  |-  ( A. x ( x  =  y  ->  ph )  -> 
( E. x  x  =  y  ->  E. x
( x  =  y  /\  ph ) ) )
31, 2mpi 17 1  |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1377   E.wex 1596
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-6 1719
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597
This theorem is referenced by:  bj-equsexv  33412  bj-equs45fv  33430  bj-equs5v  33431  bj-sb2v  33432
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