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Theorem sbi1 2119
Description: Removal of implication from substitution. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sbi1  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )

Proof of Theorem sbi1
StepHypRef Expression
1 sbequ2 1728 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ph  ->  ph ) )
2 sbequ2 1728 . . . . 5  |-  ( x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( ph  ->  ps ) ) )
31, 2syl5d 67 . . . 4  |-  ( x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  ps ) ) )
4 sbequ1 1977 . . . 4  |-  ( x  =  y  ->  ( ps  ->  [ y  /  x ] ps ) )
53, 4syl6d 69 . . 3  |-  ( x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) ) )
65sps 1851 . 2  |-  ( A. x  x  =  y  ->  ( [ y  /  x ] ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) ) )
7 sb4 2083 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ] ph  ->  A. x ( x  =  y  ->  ph )
) )
8 sb4 2083 . . . 4  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ]
( ph  ->  ps )  ->  A. x ( x  =  y  ->  ( ph  ->  ps ) ) ) )
9 ax-2 7 . . . . . 6  |-  ( ( x  =  y  -> 
( ph  ->  ps )
)  ->  ( (
x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) ) )
109al2imi 1623 . . . . 5  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  A. x
( x  =  y  ->  ps ) ) )
11 sb2 2079 . . . . 5  |-  ( A. x ( x  =  y  ->  ps )  ->  [ y  /  x ] ps )
1210, 11syl6 33 . . . 4  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) )
138, 12syl6 33 . . 3  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ]
( ph  ->  ps )  ->  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ps ) ) )
147, 13syl5d 67 . 2  |-  ( -. 
A. x  x  =  y  ->  ( [
y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) ) )
156, 14pm2.61i 164 1  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   A.wal 1381   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840  ax-13 1985
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1600  df-nf 1604  df-sb 1727
This theorem is referenced by:  spsbim  2121  sbim  2122  2sb5ndVD  33578  2sb5ndALT  33600
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