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Theorem hban 1455
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 Mario Carneiro, 2-Feb-2015.)
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 . . 3  |-  ( ph  ->  A. x ph )
2 hb.2 . . 3  |-  ( ps 
->  A. x ps )
31, 2anim12i 325 . 2  |-  ( (
ph  /\  ps )  ->  ( A. x ph  /\ 
A. x ps )
)
4 19.26 1386 . 2  |-  ( A. x ( ph  /\  ps )  <->  ( A. x ph  /\  A. x ps ) )
53, 4sylibr 141 1  |-  ( (
ph  /\  ps )  ->  A. x ( ph  /\ 
ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   A.wal 1257
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354
This theorem depends on definitions:  df-bi 114
This theorem is referenced by:  hbbi  1456  hb3an  1458  hbsbv  1833  mopick  1994  eupicka  1996  mopick2  1999  cleqh  2153
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