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Theorem hbim 1507
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 1470 . . 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 1498 . 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 1414 . 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 1312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-5 1406  ax-gen 1408  ax-4 1470  ax-i5r 1498
This theorem is referenced by:  hbbi  1510  hbia1  1514  19.21h  1519  19.38  1637  hbsbv  1892  hbmo1  2013  hbmo  2014  moexexdc  2059  2eu4  2068  cleqh  2215  hbral  2439
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