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Theorem spimt 1954
Description: Closed theorem form of spim 1955. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
Assertion
Ref Expression
spimt  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  ps ) )

Proof of Theorem spimt
StepHypRef Expression
1 ax6e 1951 . . . 4  |-  E. x  x  =  y
2 exim 1628 . . . 4  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  y  ->  E. x
( ph  ->  ps )
) )
31, 2mpi 17 . . 3  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  ->  E. x ( ph  ->  ps ) )
4 19.35 1659 . . 3  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
53, 4sylib 196 . 2  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ph  ->  E. x ps )
)
6 19.9t 1828 . . 3  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
76biimpd 207 . 2  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
85, 7sylan9r 653 1  |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps ) ) )  ->  ( A. x ph  ->  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369   A.wal 1362   E.wex 1591   F/wnf 1594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-10 1780  ax-12 1797  ax-13 1948
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-nf 1595
This theorem is referenced by: (None)
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