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Theorem bj-cbvex4vv 31389
Description: Version of cbvex4v 2137 with a dv condition, which does not require ax-13 2102. (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-cbvex4vv.1  |-  ( ( x  =  v  /\  y  =  u )  ->  ( ph  <->  ps )
)
bj-cbvex4vv.2  |-  ( ( z  =  f  /\  w  =  g )  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
bj-cbvex4vv  |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
Distinct variable groups:    z, w, ch    v, u, ph    x, y, ps    f, g, ps    z, f, g, w    w, u, x, y, z, v
Allowed substitution hints:    ph( x, y, z, w, f, g)    ps( z, w, v, u)    ch( x, y, v, u, f, g)

Proof of Theorem bj-cbvex4vv
StepHypRef Expression
1 bj-cbvex4vv.1 . . . 4  |-  ( ( x  =  v  /\  y  =  u )  ->  ( ph  <->  ps )
)
212exbidv 1781 . . 3  |-  ( ( x  =  v  /\  y  =  u )  ->  ( E. z E. w ph  <->  E. z E. w ps ) )
32bj-cbvex2vv 31386 . 2  |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. z E. w ps )
4 bj-cbvex4vv.2 . . . 4  |-  ( ( z  =  f  /\  w  =  g )  ->  ( ps  <->  ch )
)
54bj-cbvex2vv 31386 . . 3  |-  ( E. z E. w ps  <->  E. f E. g ch )
652exbii 1730 . 2  |-  ( E. v E. u E. z E. w ps  <->  E. v E. u E. f E. g ch )
73, 6bitri 257 1  |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375   E.wex 1674
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by: (None)
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