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Theorem cbvex4v 2046
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 1635 . . 3  |-  ( ( x  =  v  /\  y  =  u )  ->  ( E. z E. w ph  <->  E. z E. w ps ) )
32cbvex2v 2041 . 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 2041 . . 3  |-  ( E. z E. w ps  <->  E. f E. g ch )
652exbii 1590 . 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 1547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551
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