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Theorem hban 2014
Description: If  x is not free in  ph and  ps, it is not free in  ( ph  /\  ps ). (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Hypotheses
Ref Expression
hb.1  |-  ( ph  ->  A. x ph )
hb.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
hban  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )

Proof of Theorem hban
StepHypRef Expression
1 hb.1 . . . 4  |-  ( ph  ->  A. x ph )
21nfi 1674 . . 3  |-  F/ x ph
3 hb.2 . . . 4  |-  ( ps 
->  A. x ps )
43nfi 1674 . . 3  |-  F/ x ps
52, 4nfan 2011 . 2  |-  F/ x
( ph  /\  ps )
65nfri 1952 1  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 371   A.wal 1442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-12 1933
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1664  df-nf 1668
This theorem is referenced by:  bnj982  29590  bnj1351  29638  bnj1352  29639  bnj1441  29652  dvelimf-o  32500  ax12indalem  32516  ax12inda2ALT  32517  hbimpg  36921  hbimpgVD  37301
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