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Theorem cbvrexdva 2784
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 2297 . 2  |-  ( (
ph  /\  x  =  y )  ->  A  =  A )
31, 2cbvrexdva2 2782 1  |-  ( ph  ->  ( E. x  e.  A  ps  <->  E. y  e.  A  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wrex 2557
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535  df-cleq 2289  df-clel 2292  df-rex 2562
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