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Theorem exintrbi 1613
Description: Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
Assertion
Ref Expression
exintrbi  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
ps ) ) )

Proof of Theorem exintrbi
StepHypRef Expression
1 pm4.71 387 . . 3  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )
21albii 1447 . 2  |-  ( A. x ( ph  ->  ps )  <->  A. x ( ph  <->  (
ph  /\  ps )
) )
3 exbi 1584 . 2  |-  ( A. x ( ph  <->  ( ph  /\ 
ps ) )  -> 
( E. x ph  <->  E. x ( ph  /\  ps ) ) )
42, 3sylbi 120 1  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1330   E.wex 1469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  exintr  1614
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