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Theorem 19.43 1564
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 1429 . . . 4  |-  ( E. x ph  ->  A. x E. x ph )
2 hbe1 1429 . . . 4  |-  ( E. x ps  ->  A. x E. x ps )
31, 2hbor 1483 . . 3  |-  ( ( E. x ph  \/  E. x ps )  ->  A. x ( E. x ph  \/  E. x ps ) )
4 19.8a 1527 . . . 4  |-  ( ph  ->  E. x ph )
5 19.8a 1527 . . . 4  |-  ( ps 
->  E. x ps )
64, 5orim12i 711 . . 3  |-  ( (
ph  \/  ps )  ->  ( E. x ph  \/  E. x ps )
)
73, 6exlimih 1529 . 2  |-  ( E. x ( ph  \/  ps )  ->  ( E. x ph  \/  E. x ps ) )
8 orc 668 . . . 4  |-  ( ph  ->  ( ph  \/  ps ) )
98eximi 1536 . . 3  |-  ( E. x ph  ->  E. x
( ph  \/  ps ) )
10 olc 667 . . . 4  |-  ( ps 
->  ( ph  \/  ps ) )
1110eximi 1536 . . 3  |-  ( E. x ps  ->  E. x
( ph  \/  ps ) )
129, 11jaoi 671 . 2  |-  ( ( E. x ph  \/  E. x ps )  ->  E. x ( ph  \/  ps ) )
137, 12impbii 124 1  |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  E. x ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 664   E.wex 1426
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-io 665  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-4 1445  ax-ial 1472
This theorem depends on definitions:  df-bi 115
This theorem is referenced by:  19.44  1617  19.45  1618  19.34  1619  sborv  1818  r19.43  2525  rexun  3178  unipr  3662  uniun  3667  unopab  3909  dmun  4631  coundi  4919  coundir  4920
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