ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  spcimedv Unicode version

Theorem spcimedv 2816
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 spcimedv.2 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
21ex 114 . . 3  |-  ( ph  ->  ( x  =  A  ->  ( ch  ->  ps ) ) )
32alrimiv 1867 . 2  |-  ( ph  ->  A. x ( x  =  A  ->  ( ch  ->  ps ) ) )
4 spcimdv.1 . 2  |-  ( ph  ->  A  e.  B )
5 nfv 1521 . . 3  |-  F/ x ch
6 nfcv 2312 . . 3  |-  F/_ x A
75, 6spcimegft 2808 . 2  |-  ( A. x ( x  =  A  ->  ( ch  ->  ps ) )  -> 
( A  e.  B  ->  ( ch  ->  E. x ps ) ) )
83, 4, 7sylc 62 1  |-  ( ph  ->  ( ch  ->  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1346    = wceq 1348   E.wex 1485    e. wcel 2141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732
This theorem is referenced by:  rspcimedv  2836  fihashf1rn  10710
  Copyright terms: Public domain W3C validator