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Theorem 19.41h 1646
Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1647 instead. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.)
Hypothesis
Ref Expression
19.41h.1  |-  ( ps 
->  A. x ps )
Assertion
Ref Expression
19.41h  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )

Proof of Theorem 19.41h
StepHypRef Expression
1 19.40 1593 . . 3  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  E. x ps ) )
2 19.41h.1 . . . . 5  |-  ( ps 
->  A. x ps )
3 id 19 . . . . 5  |-  ( ps 
->  ps )
42, 3exlimih 1555 . . . 4  |-  ( E. x ps  ->  ps )
54anim2i 337 . . 3  |-  ( ( E. x ph  /\  E. x ps )  -> 
( E. x ph  /\ 
ps ) )
61, 5syl 14 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  ps ) )
7 pm3.21 262 . . . 4  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
82, 7eximdh 1573 . . 3  |-  ( ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) )
98impcom 124 . 2  |-  ( ( E. x ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
106, 9impbii 125 1  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312   E.wex 1451
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 1406  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-4 1470  ax-ial 1497
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.42h  1648  sbh  1732  sbidm  1805  19.41v  1856  2exeu  2067
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