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Theorem exintrbi 1603
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 pm4.71 611 . . 3  |-  ( (
ph  ->  ps )  <->  ( ph  <->  (
ph  /\  ps )
) )
21albii 1556 . 2  |-  ( A. x ( ph  ->  ps )  <->  A. x ( ph  <->  (
ph  /\  ps )
) )
3 exbi 1571 . 2  |-  ( A. x ( ph  <->  ( ph  /\ 
ps ) )  -> 
( E. x ph  <->  E. x ( ph  /\  ps ) ) )
42, 3sylbi 187 1  |-  ( A. x ( ph  ->  ps )  ->  ( E. x ph  <->  E. x ( ph  /\ 
ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1530   E.wex 1531
This theorem is referenced by:  exintr  1604
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532
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