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Theorem exan 1626
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 hbe1 1427 . . . 4  |-  ( E. x ph  ->  A. x E. x ph )
2119.28h 1497 . . 3  |-  ( A. x ( E. x ph  /\  ps )  <->  ( E. x ph  /\  A. x ps ) )
3 exan.1 . . 3  |-  ( E. x ph  /\  ps )
42, 3mpgbi 1384 . 2  |-  ( E. x ph  /\  A. x ps )
5 19.29r 1555 . 2  |-  ( ( E. x ph  /\  A. x ps )  ->  E. x ( ph  /\  ps ) )
64, 5ax-mp 7 1  |-  E. x
( ph  /\  ps )
Colors of variables: wff set class
Syntax hints:    /\ wa 102   A.wal 1285   E.wex 1424
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 1379  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-4 1443  ax-ial 1470
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  bm1.3ii  3935  bdbm1.3ii  11220
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