MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvex4v Structured version   Visualization version   Unicode version

Theorem cbvex4v 2137
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 1781 . . 3  |-  ( ( x  =  v  /\  y  =  u )  ->  ( E. z E. w ph  <->  E. z E. w ps ) )
32cbvex2v 2134 . 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 2134 . . 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  ax-13 2102
This theorem depends on definitions:  df-bi 190  df-an 377  df-ex 1675  df-nf 1679
This theorem is referenced by:  addsrmo  9523  mulsrmo  9524
  Copyright terms: Public domain W3C validator