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Theorem cbvex4v 1902
Description: Rule used to change bound variables, using implicit substitition. (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, z, v    x, u, w, z    y, v, w, z
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 1840 . . 3  |-  ( ( x  =  v  /\  y  =  u )  ->  ( E. z E. w ph  <->  E. z E. w ps ) )
32cbvex2v 1896 . 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 1896 . . 3  |-  ( E. z E. w ps  <->  E. f E. g ch )
652exbii 1585 . 2  |-  ( E. v E. u E. z E. w ps  <->  E. v E. u E. f E. g ch )
73, 6bitri 183 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    /\ wa 103    <-> wb 104   E.wex 1468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1437
This theorem is referenced by:  enq0sym  7247  addnq0mo  7262  mulnq0mo  7263  addsrmo  7558  mulsrmo  7559
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