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Theorem spimth 1696
Description: Closed theorem form of spim 1699. (Contributed by NM, 15-Jan-2008.) (New usage is discouraged.)
Assertion
Ref Expression
spimth  |-  ( A. x ( ( ps 
->  A. x ps )  /\  ( x  =  y  ->  ( ph  ->  ps ) ) )  -> 
( A. x ph  ->  ps ) )

Proof of Theorem spimth
StepHypRef Expression
1 imim2 55 . . . . . 6  |-  ( ( ps  ->  A. x ps )  ->  ( (
ph  ->  ps )  -> 
( ph  ->  A. x ps ) ) )
21imim2d 54 . . . . 5  |-  ( ( ps  ->  A. x ps )  ->  ( ( x  =  y  -> 
( ph  ->  ps )
)  ->  ( x  =  y  ->  ( ph  ->  A. x ps )
) ) )
32imp 123 . . . 4  |-  ( ( ( ps  ->  A. x ps )  /\  (
x  =  y  -> 
( ph  ->  ps )
) )  ->  (
x  =  y  -> 
( ph  ->  A. x ps ) ) )
43com23 78 . . 3  |-  ( ( ( ps  ->  A. x ps )  /\  (
x  =  y  -> 
( ph  ->  ps )
) )  ->  ( ph  ->  ( x  =  y  ->  A. x ps ) ) )
54al2imi 1417 . 2  |-  ( A. x ( ( ps 
->  A. x ps )  /\  ( x  =  y  ->  ( ph  ->  ps ) ) )  -> 
( A. x ph  ->  A. x ( x  =  y  ->  A. x ps ) ) )
6 ax9o 1659 . 2  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
75, 6syl6 33 1  |-  ( A. x ( ( ps 
->  A. x ps )  /\  ( x  =  y  ->  ( ph  ->  ps ) ) )  -> 
( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-i9 1493  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  equveli  1715
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