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Theorem sbi2 2242
Description: Introduction of implication into substitution. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbi2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )

Proof of Theorem sbi2
StepHypRef Expression
1 sbn 2240 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
2 pm2.21 111 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32sbimi 1811 . . 3  |-  ( [ y  /  x ]  -.  ph  ->  [ y  /  x ] ( ph  ->  ps ) )
41, 3sylbir 218 . 2  |-  ( -. 
[ y  /  x ] ph  ->  [ y  /  x ] ( ph  ->  ps ) )
5 ax-1 6 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
65sbimi 1811 . 2  |-  ( [ y  /  x ] ps  ->  [ y  /  x ] ( ph  ->  ps ) )
74, 6ja 166 1  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sbim  2244
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