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Theorem cbvex4v 1996
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypotheses
Ref Expression
cbvex4v.1  |-  ( ( x  =  v  /\  y  =  u )  ->  ( ph  <->  ps )
)
cbvex4v.2  |-  ( ( z  =  f  /\  w  =  g )  ->  ( ps  <->  ch )
)
Assertion
Ref Expression
cbvex4v  |-  ( 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    w, f    z, g    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 cbvex4v
StepHypRef Expression
1 cbvex4v.1 . . . 4  |-  ( ( x  =  v  /\  y  =  u )  ->  ( ph  <->  ps )
)
212exbidv 1638 . . 3  |-  ( ( x  =  v  /\  y  =  u )  ->  ( E. z E. w ph  <->  E. z E. w ps ) )
32cbvex2v 1993 . 2  |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. z E. w ps )
4 cbvex4v.2 . . . 4  |-  ( ( z  =  f  /\  w  =  g )  ->  ( ps  <->  ch )
)
54cbvex2v 1993 . . 3  |-  ( E. z E. w ps  <->  E. f E. g ch )
652exbii 1593 . 2  |-  ( E. v E. u E. z E. w ps  <->  E. v E. u E. f E. g ch )
73, 6bitri 241 1  |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1550
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554
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