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Theorem 19.41h 1620
Description: Theorem 19.41 of [Margaris] p. 90. New proofs should use 19.41 1621 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 1567 . . 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 1529 . . . 4  |-  ( E. x ps  ->  ps )
54anim2i 334 . . 3  |-  ( ( E. x ph  /\  E. x ps )  -> 
( E. x ph  /\ 
ps ) )
61, 5syl 14 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  ps ) )
7 pm3.21 260 . . . 4  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
82, 7eximdh 1547 . . 3  |-  ( ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) )
98impcom 123 . 2  |-  ( ( E. x ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
106, 9impbii 124 1  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426
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 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.42h  1622  sbh  1706  sbidm  1779  19.41v  1830  2exeu  2040
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