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Theorem sbcim1 3376
Description: Distribution of class substitution over implication. One direction of sbcimg 3369 that holds for proper classes. (Contributed by NM, 17-Aug-2018.)
Assertion
Ref Expression
sbcim1  |-  ( [. A  /  x ]. ( ph  ->  ps )  -> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) )

Proof of Theorem sbcim1
StepHypRef Expression
1 sbcex 3337 . 2  |-  ( [. A  /  x ]. ( ph  ->  ps )  ->  A  e.  _V )
2 sbcimg 3369 . . 3  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
32biimpd 207 . 2  |-  ( A  e.  _V  ->  ( [. A  /  x ]. ( ph  ->  ps )  ->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) ) )
41, 3mpcom 36 1  |-  ( [. A  /  x ]. ( ph  ->  ps )  -> 
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819   _Vcvv 3109   [.wsbc 3327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111  df-sbc 3328
This theorem is referenced by:  sbcimdv  3395  frege59c  38139  frege60c  38140  frege62c  38142  frege65c  38145
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