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Theorem hbbi 1481
Description: If  x is not free in  ph and  ps, it is not free in  ( ph  <->  ps ). (Contributed by NM, 5-Aug-1993.)
Hypotheses
Ref Expression
hb.1  |-  ( ph  ->  A. x ph )
hb.2  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
hbbi  |-  ( (
ph 
<->  ps )  ->  A. x
( ph  <->  ps ) )

Proof of Theorem hbbi
StepHypRef Expression
1 dfbi2 380 . 2  |-  ( (
ph 
<->  ps )  <->  ( ( ph  ->  ps )  /\  ( ps  ->  ph )
) )
2 hb.1 . . . 4  |-  ( ph  ->  A. x ph )
3 hb.2 . . . 4  |-  ( ps 
->  A. x ps )
42, 3hbim 1478 . . 3  |-  ( (
ph  ->  ps )  ->  A. x ( ph  ->  ps ) )
53, 2hbim 1478 . . 3  |-  ( ( ps  ->  ph )  ->  A. x ( ps  ->  ph ) )
64, 5hban 1480 . 2  |-  ( ( ( ph  ->  ps )  /\  ( ps  ->  ph ) )  ->  A. x
( ( ph  ->  ps )  /\  ( ps 
->  ph ) ) )
71, 6hbxfrbi 1402 1  |-  ( (
ph 
<->  ps )  ->  A. x
( ph  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1283
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-4 1441  ax-i5r 1469
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  euf  1948  sb8euh  1966
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