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Theorem reubidvag 24935
Description: Formula-building rule for restricted existential quantifier (deduction rule). (Contributed by FL, 17-Nov-2014.)
Hypotheses
Ref Expression
reubidvag.1  |-  ( ph  ->  ( ps  <->  ch )
)
reubidvag.2  |-  ( ph  ->  A  =  B )
Assertion
Ref Expression
reubidvag  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  B  ch )
)
Distinct variable groups:    x, A    x, B    ph, x
Allowed substitution hints:    ps( x)    ch( x)

Proof of Theorem reubidvag
StepHypRef Expression
1 reubidvag.2 . . 3  |-  ( ph  ->  A  =  B )
2 reueq1 2738 . . 3  |-  ( A  =  B  ->  ( E! x  e.  A  ps 
<->  E! x  e.  B  ps ) )
31, 2syl 15 . 2  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  B  ps )
)
4 reubidvag.1 . . 3  |-  ( ph  ->  ( ps  <->  ch )
)
54reubidv 2724 . 2  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! x  e.  B  ch )
)
63, 5bitrd 244 1  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623   E!wreu 2545
This theorem is referenced by:  isinob  25862  islimcat  25876  bisig0  26062  isibcg  26191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-cleq 2276  df-clel 2279  df-nfc 2408  df-reu 2550
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