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Theorem exan 1835
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 1718 . . . 4  |-  F/ x E. x ph
2119.28 1818 . . 3  |-  ( A. x ( E. x ph  /\  ps )  <->  ( E. x ph  /\  A. x ps ) )
3 exan.1 . . 3  |-  ( E. x ph  /\  ps )
42, 3mpgbi 1539 . 2  |-  ( E. x ph  /\  A. x ps )
5 19.29r 1587 . 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 1530   E.wex 1531
This theorem is referenced by:  bm1.3ii  4160  fnchoice  27803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-11 1727
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-nf 1535
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