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Theorem cbvaldva 1950
Description: Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvaldva.1  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
cbvaldva  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Distinct variable groups:    ps, y    ch, x    ph, x    ph, y
Allowed substitution hints:    ps( x)    ch( y)

Proof of Theorem cbvaldva
StepHypRef Expression
1 nfv 1605 . 2  |-  F/ y
ph
2 nfvd 1606 . 2  |-  ( ph  ->  F/ y ps )
3 cbvaldva.1 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
43ex 423 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
51, 2, 4cbvald 1948 1  |-  ( ph  ->  ( A. x ps  <->  A. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527
This theorem is referenced by:  cbvraldva2  2768
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-ex 1529  df-nf 1532
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