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Theorem 19.41 2028
Description: Theorem 19.41 of [Margaris] p. 90. See 19.41v 1823 for a version requiring fewer axioms. (Contributed by NM, 14-May-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 1725 . . 3  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  E. x ps ) )
2 19.41.1 . . . . 5  |-  F/ x ps
3219.9 1947 . . . 4  |-  ( E. x ps  <->  ps )
43anbi2i 698 . . 3  |-  ( ( E. x ph  /\  E. x ps )  <->  ( E. x ph  /\  ps )
)
51, 4sylib 199 . 2  |-  ( E. x ( ph  /\  ps )  ->  ( E. x ph  /\  ps ) )
6 pm3.21 449 . . . 4  |-  ( ps 
->  ( ph  ->  ( ph  /\  ps ) ) )
72, 6eximd 1937 . . 3  |-  ( ps 
->  ( E. x ph  ->  E. x ( ph  /\ 
ps ) ) )
87impcom 431 . 2  |-  ( ( E. x ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
95, 8impbii 190 1  |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370   E.wex 1657   F/wnf 1661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662
This theorem is referenced by:  19.42  2029  exan  2030  eean  2052  eeeanv  2054  equsexALT  2103  2sb5rf  2257  r19.41  2920  eliunxp  4934  dfopab2  6805  dfoprab3s  6806  xpcomco  7615  mpt2mptxf  28226  bnj605  29670  bnj607  29679  bj-equsexv  31252  2sb5nd  36840  2sb5ndVD  37223  2sb5ndALT  37245  eliunxp2  39718
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