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Theorem spcimedv 2995
Description: Restricted existential specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimdv.1  |-  ( ph  ->  A  e.  B )
spcimedv.2  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
Assertion
Ref Expression
spcimedv  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Distinct variable groups:    x, A    ph, x    ch, x
Allowed substitution hints:    ps( x)    B( x)

Proof of Theorem spcimedv
StepHypRef Expression
1 spcimdv.1 . . . 4  |-  ( ph  ->  A  e.  B )
2 spcimedv.2 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
32con3d 127 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  ->  -.  ch )
)
41, 3spcimdv 2993 . . 3  |-  ( ph  ->  ( A. x  -.  ps  ->  -.  ch )
)
54con2d 109 . 2  |-  ( ph  ->  ( ch  ->  -.  A. x  -.  ps )
)
6 df-ex 1548 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
75, 6syl6ibr 219 1  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359   A.wal 1546   E.wex 1547    = wceq 1649    e. wcel 1721
This theorem is referenced by:  hashf1rn  11591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918
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