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Theorem hbim 1478
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 1441 . . 3  |-  ( A. x ph  ->  ph )
2 hb.2 . . 3  |-  ( ps 
->  A. x ps )
31, 2imim12i 58 . 2  |-  ( (
ph  ->  ps )  -> 
( A. x ph  ->  A. x ps )
)
4 ax-i5r 1469 . 2  |-  ( ( A. x ph  ->  A. x ps )  ->  A. x ( A. x ph  ->  ps ) )
5 hb.1 . . . 4  |-  ( ph  ->  A. x ph )
65imim1i 59 . . 3  |-  ( ( A. x ph  ->  ps )  ->  ( ph  ->  ps ) )
76alimi 1385 . 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 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-5 1377  ax-gen 1379  ax-4 1441  ax-i5r 1469
This theorem is referenced by:  hbbi  1481  hbia1  1485  19.21h  1490  19.38  1607  hbsbv  1860  hbmo1  1981  hbmo  1982  moexexdc  2027  2eu4  2036  cleqh  2182  hbral  2401
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