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Theorem sbi2 2004
Description: Introduction of implication into substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbi2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )

Proof of Theorem sbi2
StepHypRef Expression
1 sbn 2002 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
2 pm2.21 100 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32sbimi 1633 . . 3  |-  ( [ y  /  x ]  -.  ph  ->  [ y  /  x ] ( ph  ->  ps ) )
41, 3sylbir 204 . 2  |-  ( -. 
[ y  /  x ] ph  ->  [ y  /  x ] ( ph  ->  ps ) )
5 ax-1 5 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
65sbimi 1633 . 2  |-  ( [ y  /  x ] ps  ->  [ y  /  x ] ( ph  ->  ps ) )
74, 6ja 153 1  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   [wsb 1629
This theorem is referenced by:  sbim  2005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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