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Theorem 19.41 1665
Description: Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 12-Jan-2018.)
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 1611 . . 3  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  E. x ps ) )
2 19.41.1 . . . . 5  |-  F/ x ps
3219.9 1624 . . . 4  |-  ( E. x ps  <->  ps )
43anbi2i 453 . . 3  |-  ( ( E. x ph  /\  E. x ps )  <->  ( E. x ph  /\  ps )
)
51, 4sylib 121 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  ps ) )
6 pm3.21 262 . . . 4  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
72, 6eximd 1592 . . 3  |-  ( ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) )
87impcom 124 . 2  |-  ( ( E. x ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
95, 8impbii 125 1  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    <-> wb 104   F/wnf 1437   E.wex 1469
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 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515
This theorem depends on definitions:  df-bi 116  df-nf 1438
This theorem is referenced by:  19.42  1667  eean  1904  r19.41  2589  eliunxp  4685  dfopab2  6094  dfoprab3s  6095  xpcomco  6727
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