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Theorem spsbim 1768
Description: Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
Assertion
Ref Expression
spsbim  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )

Proof of Theorem spsbim
StepHypRef Expression
1 imim2 54 . . . 4  |-  ( (
ph  ->  ps )  -> 
( ( x  =  y  ->  ph )  -> 
( x  =  y  ->  ps ) ) )
21sps 1473 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( (
x  =  y  ->  ph )  ->  ( x  =  y  ->  ps ) ) )
3 id 19 . . . . . 6  |-  ( (
ph  ->  ps )  -> 
( ph  ->  ps )
)
43anim2d 330 . . . . 5  |-  ( (
ph  ->  ps )  -> 
( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  ps ) ) )
54alimi 1387 . . . 4  |-  ( A. x ( ph  ->  ps )  ->  A. x
( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  ps ) ) )
6 exim 1533 . . . 4  |-  ( A. x ( ( x  =  y  /\  ph )  ->  ( x  =  y  /\  ps )
)  ->  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ps ) ) )
75, 6syl 14 . . 3  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ( x  =  y  /\  ph )  ->  E. x ( x  =  y  /\  ps ) ) )
82, 7anim12d 328 . 2  |-  ( A. x ( ph  ->  ps )  ->  ( (
( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
)  ->  ( (
x  =  y  ->  ps )  /\  E. x
( x  =  y  /\  ps ) ) ) )
9 df-sb 1690 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  ph )  /\  E. x ( x  =  y  /\  ph )
) )
10 df-sb 1690 . 2  |-  ( [ y  /  x ] ps 
<->  ( ( x  =  y  ->  ps )  /\  E. x ( x  =  y  /\  ps ) ) )
118, 9, 103imtr4g 203 1  |-  ( A. x ( ph  ->  ps )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1285   E.wex 1424   [wsb 1689
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115  df-sb 1690
This theorem is referenced by:  spsbbi  1769  hbsb4t  1934  moim  2009
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