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Theorem sbc19.21g 3005
Description: Substitution for a variable not free in antecedent affects only the consequent. (Contributed by NM, 11-Oct-2004.)
Hypothesis
Ref Expression
sbcgf.1  |-  F/ x ph
Assertion
Ref Expression
sbc19.21g  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )

Proof of Theorem sbc19.21g
StepHypRef Expression
1 sbcimg 2978 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
2 sbcgf.1 . . . 4  |-  F/ x ph
32sbcgf 3004 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
43imbi1d 230 . 2  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
51, 4bitrd 187 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   F/wnf 1440    e. wcel 2128   [.wsbc 2937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-sbc 2938
This theorem is referenced by: (None)
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