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Theorem cbvreuv 2592
Description: Change the bound variable of a restricted unique existential quantifier using implicit substitution. (Contributed by NM, 5-Apr-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
cbvralv.1  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
cbvreuv  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Distinct variable groups:    x, A    y, A    ph, y    ps, x
Allowed substitution hints:    ph( x)    ps( y)

Proof of Theorem cbvreuv
StepHypRef Expression
1 nfv 1466 . 2  |-  F/ y
ph
2 nfv 1466 . 2  |-  F/ x ps
3 cbvralv.1 . 2  |-  ( x  =  y  ->  ( ph 
<->  ps ) )
41, 2, 3cbvreu 2588 1  |-  ( E! x  e.  A  ph  <->  E! y  e.  A  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   E!wreu 2361
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-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-cleq 2081  df-clel 2084  df-reu 2366
This theorem is referenced by:  reu8  2811
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