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Theorem sbi1 2016
Description: Removal of implication from substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbi1  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )

Proof of Theorem sbi1
StepHypRef Expression
1 sbequ2 1640 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
2 sbequ2 1640 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( ph  ->  ps ) ) )
31, 2syl5d 62 . . . 4  |-  ( x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  ps ) ) )
4 sbequ1 1871 . . . 4  |-  ( x  =  y  ->  ( ps  ->  [ y  /  x ] ps ) )
53, 4syl6d 64 . . 3  |-  ( x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) ) )
65sps 1751 . 2  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) ) )
7 sb4 2006 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
8 sb4 2006 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ]
( ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) )
9 ax-2 6 . . . . . 6  |-  ( ( x  =  y  -> 
( ph  ->  ps )
)  ->  ( (
x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) ) )
109al2imi 1551 . . . . 5  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  A. x
( x  =  y  ->  ps ) ) )
11 sb2 1976 . . . . 5  |-  ( A. x ( x  =  y  ->  ps )  ->  [ y  /  x ] ps )
1210, 11syl6 29 . . . 4  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) )
138, 12syl6 29 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ]
( ph  ->  ps )  ->  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) ) )
147, 13syl5d 62 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) ) )
156, 14pm2.61i 156 1  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1530   [wsb 1638
This theorem is referenced by:  sbim  2018  spsbim  2029  2sb5ndVD  29002  2sb5ndALT  29025
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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