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Theorem cbvrexdva 2589
Description: Rule used to change the bound variable in a restricted existential 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
cbvrexdva  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. 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 cbvrexdva
StepHypRef Expression
1 cbvraldva.1 . 2  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
2 eqidd 2084 . 2  |-  ( (
ph  /\  x  =  y )  ->  A  =  A )
31, 2cbvrexdva2 2587 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. y  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   E.wrex 2354
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-cleq 2076  df-clel 2079  df-rex 2359
This theorem is referenced by:  tfrlem3ag  5978  tfrlem3a  5979  tfrlemi1  6001  tfr1onlem3ag  6006
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