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

Theorem spim 1717
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1717 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
Hypotheses
Ref Expression
spim.1  |-  F/ x ps
spim.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spim  |-  ( A. x ph  ->  ps )

Proof of Theorem spim
StepHypRef Expression
1 spim.1 . . 3  |-  F/ x ps
21nfri 1500 . 2  |-  ( ps 
->  A. x ps )
3 spim.2 . 2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
42, 3spimh 1716 1  |-  ( A. x ph  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330   F/wnf 1437
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-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-i9 1511  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  cbv3  1721  chvar  1731  spimv  1784  2spim  13137
  Copyright terms: Public domain W3C validator