MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spcimedv Unicode version

Theorem spcimedv 2943
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 125 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  ->  -.  ch )
)
41, 3spcimdv 2941 . . 3  |-  ( ph  ->  ( A. x  -.  ps  ->  -.  ch )
)
54con2d 107 . 2  |-  ( ph  ->  ( ch  ->  -.  A. x  -.  ps )
)
6 df-ex 1542 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
75, 6syl6ibr 218 1  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1540   E.wex 1541    = wceq 1642    e. wcel 1710
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866
  Copyright terms: Public domain W3C validator