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Theorem cbvraldva 2637
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
cbvraldva.1  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
cbvraldva  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. y  e.  A  ch )
)
Distinct variable groups:    ps, y    ch, x    x, A, y    ph, x, y
Allowed substitution hints:    ps( x)    ch( y)

Proof of Theorem cbvraldva
StepHypRef Expression
1 cbvraldva.1 . 2  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
2 eqidd 2118 . 2  |-  ( (
ph  /\  x  =  y )  ->  A  =  A )
31, 2cbvraldva2 2635 1  |-  ( ph  ->  ( A. x  e.  A  ps  <->  A. y  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wral 2393
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 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-cleq 2110  df-clel 2113  df-ral 2398
This theorem is referenced by: (None)
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