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Theorem spimt 1671
Description: Closed theorem form of spim 1673. (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 1631 . . . 4  |-  E. x  x  =  y
2 exim 1535 . . . 4  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( E. x  x  =  y  ->  E. x
( ph  ->  ps )
) )
31, 2mpi 15 . . 3  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  ->  E. x ( ph  ->  ps ) )
4 19.35-1 1560 . . 3  |-  ( E. x ( ph  ->  ps )  ->  ( A. x ph  ->  E. x ps ) )
53, 4syl 14 . 2  |-  ( A. x ( x  =  y  ->  ( ph  ->  ps ) )  -> 
( A. x ph  ->  E. x ps )
)
6 19.9t 1578 . . 3  |-  ( F/ x ps  ->  ( E. x ps  <->  ps )
)
76biimpd 142 . 2  |-  ( F/ x ps  ->  ( E. x ps  ->  ps ) )
85, 7sylan9r 402 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 102   A.wal 1287    = wceq 1289   F/wnf 1394   E.wex 1426
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-i9 1468  ax-ial 1472
This theorem depends on definitions:  df-bi 115  df-nf 1395
This theorem is referenced by:  spimd  11312
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