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Theorem sbi2 2017
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 2015 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
2 pm2.21 100 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32sbimi 1642 . . 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 1642 . 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 1638
This theorem is referenced by:  sbim  2018
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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