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Theorem r19.45zv 3725
Description: Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
Assertion
Ref Expression
r19.45zv  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  E. x  e.  A  ps ) ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hint:    ps( x)

Proof of Theorem r19.45zv
StepHypRef Expression
1 r19.9rzv 3722 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  E. x  e.  A  ph ) )
21orbi1d 684 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  \/  E. x  e.  A  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) ) )
3 r19.43 2863 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
42, 3syl6rbbr 256 1  |-  ( A  =/=  (/)  ->  ( E. x  e.  A  ( ph  \/  ps )  <->  ( ph  \/  E. x  e.  A  ps ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    =/= wne 2599   E.wrex 2706   (/)c0 3628
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-nul 3629
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