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Theorem cbval2v 1946
Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
Hypothesis
Ref Expression
cbval2v.1  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
Assertion
Ref Expression
cbval2v  |-  ( A. x A. y ph  <->  A. z A. 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 cbval2v
StepHypRef Expression
1 nfv 1605 . 2  |-  F/ z
ph
2 nfv 1605 . 2  |-  F/ w ph
3 nfv 1605 . 2  |-  F/ x ps
4 nfv 1605 . 2  |-  F/ y ps
5 cbval2v.1 . 2  |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph  <->  ps )
)
61, 2, 3, 4, 5cbval2 1944 1  |-  ( A. x A. y ph  <->  A. z A. w ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527
This theorem is referenced by:  seqf1o  11087
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532
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