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Theorem sbi2 2137
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 2134 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
2 pm2.21 103 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32sbimi 1666 . . 3  |-  ( [ y  /  x ]  -.  ph  ->  [ y  /  x ] ( ph  ->  ps ) )
41, 3sylbir 206 . 2  |-  ( -. 
[ y  /  x ] ph  ->  [ y  /  x ] ( ph  ->  ps ) )
5 ax-1 6 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
65sbimi 1666 . 2  |-  ( [ y  /  x ] ps  ->  [ y  /  x ] ( ph  ->  ps ) )
74, 6ja 156 1  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   [wsb 1659
This theorem is referenced by:  sbim  2139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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