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

Theorem spimeh 1702
Description: Existential introduction, using implicit substitition. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by NM, 3-Feb-2015.) (New usage is discouraged.)
Hypotheses
Ref Expression
spimeh.1  |-  ( ph  ->  A. x ph )
spimeh.2  |-  ( x  =  y  ->  ( ph  ->  ps ) )
Assertion
Ref Expression
spimeh  |-  ( ph  ->  E. x ps )

Proof of Theorem spimeh
StepHypRef Expression
1 a9e 1659 . 2  |-  E. x  x  =  y
2 spimeh.1 . . 3  |-  ( ph  ->  A. x ph )
3 spimeh.2 . . . 4  |-  ( x  =  y  ->  ( ph  ->  ps ) )
43com12 30 . . 3  |-  ( ph  ->  ( x  =  y  ->  ps ) )
52, 4eximdh 1575 . 2  |-  ( ph  ->  ( E. x  x  =  y  ->  E. x ps ) )
61, 5mpi 15 1  |-  ( ph  ->  E. x ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1314    = wceq 1316   E.wex 1453
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 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator