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Theorem spimt 1955
Description: Closed theorem form of spim 1957. (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 a9e 1952 . . . 4  |-  E. x  x  =  y
2 exim 1584 . . . 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 1610 . . 3  |-  ( E. x ( ph  ->  ps )  <->  ( A. x ph  ->  E. x ps )
)
53, 4sylib 189 . 2  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ph  ->  E. x ps )
)
6 19.9t 1793 . . 3  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
76biimpd 199 . 2  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
85, 7sylan9r 640 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 359   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  spimOLD  1958  equveliOLD  2086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
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