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Theorem 19.41 1799
Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
Hypothesis
Ref Expression
19.41.1  |-  F/ x ps
Assertion
Ref Expression
19.41  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )

Proof of Theorem 19.41
StepHypRef Expression
1 19.40 1608 . . 3  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  E. x ps ) )
2 19.41.1 . . . . 5  |-  F/ x ps
3 id 21 . . . . 5  |-  ( ps 
->  ps )
42, 3exlimi 1781 . . . 4  |-  ( E. x ps  ->  ps )
54anim2i 555 . . 3  |-  ( ( E. x ph  /\  E. x ps )  -> 
( E. x ph  /\ 
ps ) )
61, 5syl 17 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  ps ) )
7 pm3.21 437 . . . 4  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
82, 7eximd 1711 . . 3  |-  ( ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) )
98impcom 421 . 2  |-  ( ( E. x ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
106, 9impbii 182 1  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1537   F/wnf 1539
This theorem is referenced by:  19.42  1800  19.41v  2035  eean  2057  r19.41  2667  eliunxp  4811  dfopab2  6108  dfoprab3s  6109  xpcomco  6920  2sb5nd  27379  2sb5ndVD  27736  2sb5ndALT  27759  bnj605  27988  bnj607  27997
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-gen 1536  ax-4 1692
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1538  df-nf 1540
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