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Theorem 19.41 1619
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 1565 . . 3  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  E. x ps ) )
2 19.41.1 . . . . 5  |-  F/ x ps
3219.9 1578 . . . 4  |-  ( E. x ps  <->  ps )
43anbi2i 445 . . 3  |-  ( ( E. x ph  /\  E. x ps )  <->  ( E. x ph  /\  ps )
)
51, 4sylib 120 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  ps ) )
6 pm3.21 260 . . . 4  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
72, 6eximd 1546 . . 3  |-  ( ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) )
87impcom 123 . 2  |-  ( ( E. x ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
95, 8impbii 124 1  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    <-> wb 103   F/wnf 1392   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  df-nf 1393
This theorem is referenced by:  19.42  1621  eean  1851  r19.41  2518  eliunxp  4543  dfopab2  5916  dfoprab3s  5917  xpcomco  6494
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