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Theorem spimt 1946
Description: Closed theorem form of spim 1947. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.)
Assertion
Ref Expression
spimt  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  ps ) )

Proof of Theorem spimt
StepHypRef Expression
1 nfnf1 1780 . . . . 5  |-  F/ x F/ x ps
2 nfa1 1779 . . . . 5  |-  F/ x A. x ph
31, 2nfan 1800 . . . 4  |-  F/ x
( F/ x ps 
/\  A. x ph )
4 sp 1733 . . . . . . 7  |-  ( A. x ph  ->  ph )
54adantl 452 . . . . . 6  |-  ( ( F/ x ps  /\  A. x ph )  ->  ph )
6 nfr 1765 . . . . . . 7  |-  ( F/ x ps  ->  ( ps  ->  A. x ps )
)
76adantr 451 . . . . . 6  |-  ( ( F/ x ps  /\  A. x ph )  -> 
( ps  ->  A. x ps ) )
85, 7embantd 50 . . . . 5  |-  ( ( F/ x ps  /\  A. x ph )  -> 
( ( ph  ->  ps )  ->  A. x ps ) )
98imim2d 48 . . . 4  |-  ( ( F/ x ps  /\  A. x ph )  -> 
( ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( x  =  y  ->  A. x ps )
) )
103, 9alimd 1768 . . 3  |-  ( ( F/ x ps  /\  A. x ph )  -> 
( A. x ( x  =  y  -> 
( ph  ->  ps )
)  ->  A. x
( x  =  y  ->  A. x ps )
) )
1110impancom 427 . 2  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  A. x ( x  =  y  ->  A. x ps ) ) )
12 ax9o 1922 . 2  |-  ( A. x ( x  =  y  ->  A. x ps )  ->  ps )
1311, 12syl6 29 1  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1531   F/wnf 1535
This theorem is referenced by:  spim  1947  equveli  1960  a12lem1  28948
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1533  df-nf 1536
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