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Theorem sbi1v 1787
Description: Forward direction of sbimv 1789. (Contributed by Jim Kingdon, 25-Dec-2017.)
Assertion
Ref Expression
sbi1v  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    ps( x, y)

Proof of Theorem sbi1v
StepHypRef Expression
1 sb6 1782 . 2  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
2 sb6 1782 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )
3 ax-2 6 . . . . 5  |-  ( ( x  =  y  -> 
( ph  ->  ps )
)  ->  ( (
x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) ) )
43al2imi 1363 . . . 4  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  A. x
( x  =  y  ->  ps ) ) )
5 sb2 1666 . . . 4  |-  ( A. x ( x  =  y  ->  ps )  ->  [ y  /  x ] ps )
64, 5syl6 33 . . 3  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) )
72, 6sylbi 118 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) )
81, 7syl5bi 145 1  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257   [wsb 1661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-11 1413  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-sb 1662
This theorem is referenced by:  sbimv  1789
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