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Theorem equsexhv 2086
Description: Version of equsexh 2145 with a dv condition, which does not require ax-13 2104. (Contributed by BJ, 31-May-2019.)
Hypotheses
Ref Expression
equsexhv.nf  |-  ( ps 
->  A. x ps )
equsexhv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
equsexhv  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem equsexhv
StepHypRef Expression
1 equsexhv.nf . . 3  |-  ( ps 
->  A. x ps )
21nfi 1682 . 2  |-  F/ x ps
3 equsexhv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
42, 3equsexv 2085 1  |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676
This theorem is referenced by:  cleljustALT  2102
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