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Theorem hban 1833
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 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 1551 . . 3  |-  F/ x ph
3 hb.2 . . . 4  |-  ( ps 
->  A. x ps )
43nfi 1551 . . 3  |-  F/ x ps
52, 4nfan 1829 . 2  |-  F/ x
( ph  /\  ps )
65nfri 1763 1  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358   A.wal 1540
This theorem is referenced by:  hb3anOLD  1836  dvelimv  1944  dvelimh  1969  dvelimALT  2138  dvelimf-o  2185  ax11indalem  2202  ax11inda2ALT  2203  cleqh  2455  hbimpg  28065  hbimpgVD  28442  bnj982  28572  bnj1351  28621  bnj1352  28622  bnj1441  28635  dvelimvNEW7  28885  dvelimhvAUX7  28915  dvelimALTNEW7  29054  dvelimhOLD7  29114  ax12-2  29172  ax10lem17ALT  29192  a12studyALT  29202
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545
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