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Theorem 19.43 1607
Description: Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 2-Feb-2015.)
Assertion
Ref Expression
19.43  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )

Proof of Theorem 19.43
StepHypRef Expression
1 hbe1 1471 . . . 4  |-  ( E. x ph  ->  A. x E. x ph )
2 hbe1 1471 . . . 4  |-  ( E. x ps  ->  A. x E. x ps )
31, 2hbor 1525 . . 3  |-  ( ( E. x ph  \/  E. x ps )  ->  A. x ( E. x ph  \/  E. x ps ) )
4 19.8a 1569 . . . 4  |-  ( ph  ->  E. x ph )
5 19.8a 1569 . . . 4  |-  ( ps 
->  E. x ps )
64, 5orim12i 748 . . 3  |-  ( (
ph  \/  ps )  ->  ( E. x ph  \/  E. x ps )
)
73, 6exlimih 1572 . 2  |-  ( E. x ( ph  \/  ps )  ->  ( E. x ph  \/  E. x ps ) )
8 orc 701 . . . 4  |-  ( ph  ->  ( ph  \/  ps ) )
98eximi 1579 . . 3  |-  ( E. x ph  ->  E. x
( ph  \/  ps ) )
10 olc 700 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
1110eximi 1579 . . 3  |-  ( E. x ps  ->  E. x
( ph  \/  ps ) )
129, 11jaoi 705 . 2  |-  ( ( E. x ph  \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
137, 12impbii 125 1  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 697   E.wex 1468
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-io 698  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-ial 1514
This theorem depends on definitions:  df-bi 116
This theorem is referenced by:  19.44  1660  19.45  1661  19.34  1662  sborv  1862  r19.43  2589  rexun  3256  unipr  3750  uniun  3755  unopab  4007  dmun  4746  coundi  5040  coundir  5041
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