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Theorem dedths2 32338
Description: Generalization of dedths 32335 that is not useful unless we can separately prove  |-  A  e.  _V. (Contributed by NM, 13-Jun-2019.)
Hypothesis
Ref Expression
dedths2.1  |-  [. if ( [. A  /  x ]. ph ,  A ,  B )  /  x ]. ps
Assertion
Ref Expression
dedths2  |-  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )

Proof of Theorem dedths2
StepHypRef Expression
1 dfsbcq 3185 . 2  |-  ( A  =  if ( [. A  /  x ]. ph ,  A ,  B )  ->  ( [. A  /  x ]. ps  <->  [. if (
[. A  /  x ]. ph ,  A ,  B )  /  x ]. ps ) )
2 dedths2.1 . 2  |-  [. if ( [. A  /  x ]. ph ,  A ,  B )  /  x ]. ps
31, 2dedth 3838 1  |-  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   [.wsbc 3183   ifcif 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-clab 2428  df-cleq 2434  df-clel 2437  df-sbc 3184  df-if 3789
This theorem is referenced by: (None)
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