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Theorem equsexv 2076
Description: Version of equsex 2140 with a dv condition, which does not require ax-13 2101. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsexv.nf  |-  F/ x ps
equsexv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsexv  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem equsexv
StepHypRef Expression
1 equsexv.1 . . . 4  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
21pm5.32i 647 . . 3  |-  ( ( x  =  y  /\  ph )  <->  ( x  =  y  /\  ps )
)
32exbii 1728 . 2  |-  ( E. x ( x  =  y  /\  ph )  <->  E. x ( x  =  y  /\  ps )
)
4 ax6ev 1817 . . 3  |-  E. x  x  =  y
5 equsexv.nf . . . 4  |-  F/ x ps
6519.41 2061 . . 3  |-  ( E. x ( x  =  y  /\  ps )  <->  ( E. x  x  =  y  /\  ps )
)
74, 6mpbiran 934 . 2  |-  ( E. x ( x  =  y  /\  ps )  <->  ps )
83, 7bitri 257 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   E.wex 1673   F/wnf 1677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1674  df-nf 1678
This theorem is referenced by:  equsexhv  2077  sb56  2091  cleljustALT2  2100  sb10f  2295  dprd2d2  17725  poimirlem25  32009
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