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Theorem sbc19.21g 3389
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 3366 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )
) )
2 sbcgf.1 . . . 4  |-  F/ x ph
32sbcgf 3388 . . 3  |-  ( A  e.  V  ->  ( [. A  /  x ]. ph  <->  ph ) )
43imbi1d 315 . 2  |-  ( A  e.  V  ->  (
( [. A  /  x ]. ph  ->  [. A  /  x ]. ps )  <->  ( ph  ->  [. A  /  x ]. ps ) ) )
51, 4bitrd 253 1  |-  ( A  e.  V  ->  ( [. A  /  x ]. ( ph  ->  ps ) 
<->  ( ph  ->  [. A  /  x ]. ps )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184   F/wnf 1621    e. wcel 1823   [.wsbc 3324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-v 3108  df-sbc 3325
This theorem is referenced by:  bnj121  34329  bnj124  34330  bnj130  34333  bnj207  34340  bnj611  34377  bnj1000  34400
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