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Theorem sbi2 1957
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 1955 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
2 pm2.21 102 . . . 4  |-  ( -. 
ph  ->  ( ph  ->  ps ) )
32sbimi 1885 . . 3  |-  ( [ y  /  x ]  -.  ph  ->  [ y  /  x ] ( ph  ->  ps ) )
41, 3sylbir 206 . 2  |-  ( -. 
[ y  /  x ] ph  ->  [ y  /  x ] ( ph  ->  ps ) )
5 ax-1 7 . . 3  |-  ( ps 
->  ( ph  ->  ps ) )
65sbimi 1885 . 2  |-  ( [ y  /  x ] ps  ->  [ y  /  x ] ( ph  ->  ps ) )
74, 6ja 155 1  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  [ y  /  x ] ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   [wsb 1883
This theorem is referenced by:  sbim  1958
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884
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