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Theorem cbvex2v 1991
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
Hypothesis
Ref Expression
cbval2v.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbvex2v  |-  ( E. x E. y ph  <->  E. z E. w ps )
Distinct variable groups:    z, w, ph    x, y, ps    x, w    y, z
Allowed substitution hints:    ph( x, y)    ps( z, w)

Proof of Theorem cbvex2v
StepHypRef Expression
1 nfv 1674 . 2  |-  F/ z
ph
2 nfv 1674 . 2  |-  F/ w ph
3 nfv 1674 . 2  |-  F/ x ps
4 nfv 1674 . 2  |-  F/ y ps
5 cbval2v.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbvex2 1988 1  |-  ( E. x E. y ph  <->  E. z E. w ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369   E.wex 1587
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591
This theorem is referenced by:  cbvex4v  1994  2moOLDOLD  2371  2eu6OLD  2381  th3qlem1  7315
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