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Theorem cbvexdva 1853
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 1467 . 2  |-  F/ y
ph
2 nfvd 1468 . 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 1851 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   E.wex 1427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473
This theorem depends on definitions:  df-bi 116  df-nf 1396
This theorem is referenced by:  cbvrexdva2  2596  acexmid  5665  tfrlemi1  6111  ltexpri  7226  recexpr  7251
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