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Theorem hban 1736
Description: If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
hban.1  |-  ( ph  ->  A. x ph )
hban.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
hban  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )

Proof of Theorem hban
StepHypRef Expression
1 df-an 360 . 2  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
2 hban.1 . . . 4  |-  ( ph  ->  A. x ph )
3 hban.2 . . . . 5  |-  ( ps 
->  A. x ps )
43hbn 1720 . . . 4  |-  ( -. 
ps  ->  A. x  -.  ps )
52, 4hbim 1725 . . 3  |-  ( (
ph  ->  -.  ps )  ->  A. x ( ph  ->  -.  ps ) )
65hbn 1720 . 2  |-  ( -.  ( ph  ->  -.  ps )  ->  A. x  -.  ( ph  ->  -.  ps ) )
71, 6hbxfrbi 1555 1  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   A.wal 1527
This theorem is referenced by:  hb3an  1759  dvelimv  1879  dvelimh  1904  dvelimALT  2072  dvelimf-o  2119  ax11indalem  2136  ax11inda2ALT  2137  cleqh  2380  hbimpg  28320  hbimpgVD  28680  bnj982  28810  bnj1351  28859  bnj1352  28860  bnj1441  28873  ax12-2  29103  ax10lem17ALT  29123  a12studyALT  29133
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360
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