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Theorem hbim 1826
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.) (Proof shortened by Wolf Lammen, 1-Jan-2018.)
Hypotheses
Ref Expression
hbim.1  |-  ( ph  ->  A. x ph )
hbim.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
hbim  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )

Proof of Theorem hbim
StepHypRef Expression
1 hbim.1 . 2  |-  ( ph  ->  A. x ph )
2 hbim.2 . . 3  |-  ( ps 
->  A. x ps )
32a1i 11 . 2  |-  ( ph  ->  ( ps  ->  A. x ps ) )
41, 3hbim1 1819 1  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546
This theorem is referenced by:  19.23hOLD  1829  hbanOLD  1841  19.21hOLD  1852  cbv3hvOLD  1863  ax12olem5OLD  1971  axi5r  2361  cleqh  2485  hbral  2698  cbv3hvNEW7  28785
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-11 1753
This theorem depends on definitions:  df-bi 178  df-ex 1548  df-nf 1551
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