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

Theorem hbim 1556
Description: If  x is not free in  ph and  ps, it is not free in  ( ph  ->  ps ). (Contributed by NM, 5-Aug-1993.) (Proof shortened by O'Cat, 3-Mar-2008.) (Revised by Mario Carneiro, 2-Feb-2015.)
Hypotheses
Ref Expression
hb.1  |-  ( ph  ->  A. x ph )
hb.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
hbim  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )

Proof of Theorem hbim
StepHypRef Expression
1 ax-4 1521 . . 3  |-  ( A. x ph  ->  ph )
2 hb.2 . . 3  |-  ( ps 
->  A. x ps )
31, 2imim12i 59 . 2  |-  ( (
ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
4 ax-i5r 1546 . 2  |-  ( ( A. x ph  ->  A. x ps )  ->  A. x ( A. x ph  ->  ps ) )
5 hb.1 . . . 4  |-  ( ph  ->  A. x ph )
65imim1i 60 . . 3  |-  ( ( A. x ph  ->  ps )  ->  ( ph  ->  ps ) )
76alimi 1466 . 2  |-  ( A. x ( A. x ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
83, 4, 73syl 17 1  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1362
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1458  ax-gen 1460  ax-4 1521  ax-i5r 1546
This theorem is referenced by:  hbbi  1559  hbia1  1563  19.21h  1568  19.38  1687  hbsbv  1953  hbmo1  2076  hbmo  2077  moexexdc  2122  2eu4  2131  cleqh  2289  hbral  2519
  Copyright terms: Public domain W3C validator