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Theorem dedths2 34169
Description: Generalization of dedths 34166 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 3338 . 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 3997 1  |-  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   [.wsbc 3336   ifcif 3945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-sbc 3337  df-if 3946
This theorem is referenced by: (None)
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