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Theorem hb3an 1835
Description: If  x is not free in  ph,  ps, and  ch, it is not free in  ( ph  /\  ps  /\  ch ). (Contributed by NM, 14-Sep-2003.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
Hypotheses
Ref Expression
hb.1  |-  ( ph  ->  A. x ph )
hb.2  |-  ( ps 
->  A. x ps )
hb.3  |-  ( ch 
->  A. x ch )
Assertion
Ref Expression
hb3an  |-  ( (
ph  /\  ps  /\  ch )  ->  A. x ( ph  /\ 
ps  /\  ch )
)

Proof of Theorem hb3an
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
5 hb.3 . . . 4  |-  ( ch 
->  A. x ch )
65nfi 1551 . . 3  |-  F/ x ch
72, 4, 6nf3an 1832 . 2  |-  F/ x
( ph  /\  ps  /\  ch )
87nfri 1763 1  |-  ( (
ph  /\  ps  /\  ch )  ->  A. x ( ph  /\ 
ps  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934   A.wal 1540
This theorem is referenced by:  bnj982  28572  bnj1276  28609  bnj1350  28620
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-3an 936  df-ex 1542  df-nf 1545
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