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Theorem exan 1823
Description: Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
exan.1  |-  ( E. x ph  /\  ps )
Assertion
Ref Expression
exan  |-  E. x
( ph  /\  ps )

Proof of Theorem exan
StepHypRef Expression
1 nfe1 1706 . . . 4  |-  F/ x E. x ph
2119.28 1806 . . 3  |-  ( A. x ( E. x ph  /\  ps )  <->  ( E. x ph  /\  A. x ps ) )
3 exan.1 . . 3  |-  ( E. x ph  /\  ps )
42, 3mpgbi 1536 . 2  |-  ( E. x ph  /\  A. x ps )
5 19.29r 1584 . 2  |-  ( ( E. x ph  /\  A. x ps )  ->  E. x ( ph  /\  ps ) )
64, 5ax-mp 8 1  |-  E. x
( ph  /\  ps )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   A.wal 1527   E.wex 1528
This theorem is referenced by:  bm1.3ii  4144  fnchoice  27700
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-11 1715
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532
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