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Theorem cbvexdva 1922
Description: Rule used to change the bound variable in an existential 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
cbvexdva  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Distinct variable groups:    ps, y    ch, x    ph, x    ph, y
Allowed substitution hints:    ps( x)    ch( y)

Proof of Theorem cbvexdva
StepHypRef Expression
1 nfv 1521 . 2  |-  F/ y
ph
2 nfvd 1522 . 2  |-  ( ph  ->  F/ y ps )
3 cbvaldva.1 . . 3  |-  ( (
ph  /\  x  =  y )  ->  ( ps 
<->  ch ) )
43ex 114 . 2  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
51, 2, 4cbvexd 1920 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1485
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 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527
This theorem depends on definitions:  df-bi 116  df-nf 1454
This theorem is referenced by:  cbvrexdva2  2704  acexmid  5850  tfrlemi1  6309  ltexpri  7564  recexpr  7589
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