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Theorem hb3an 1771
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.)
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 df-3an 936 . 2  |-  ( (
ph  /\  ps  /\  ch ) 
<->  ( ( ph  /\  ps )  /\  ch )
)
2 hb.1 . . . 4  |-  ( ph  ->  A. x ph )
3 hb.2 . . . 4  |-  ( ps 
->  A. x ps )
42, 3hban 1748 . . 3  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
5 hb.3 . . 3  |-  ( ch 
->  A. x ch )
64, 5hban 1748 . 2  |-  ( ( ( ph  /\  ps )  /\  ch )  ->  A. x ( ( ph  /\ 
ps )  /\  ch ) )
71, 6hbxfrbi 1558 1  |-  ( (
ph  /\  ps  /\  ch )  ->  A. x ( ph  /\ 
ps  /\  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934   A.wal 1530
This theorem is referenced by:  bnj982  29126  bnj1276  29163  bnj1350  29174
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-3an 936
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