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Theorem r19.45zv 3552
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 3549 . . 3  |-  ( A  =/=  (/)  ->  ( ph  <->  E. x  e.  A  ph ) )
21orbi1d 683 . 2  |-  ( A  =/=  (/)  ->  ( ( ph  \/  E. x  e.  A  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) ) )
3 r19.43 2696 . 2  |-  ( E. x  e.  A  (
ph  \/  ps )  <->  ( E. x  e.  A  ph  \/  E. x  e.  A  ps ) )
42, 3syl6rbbr 255 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 176    \/ wo 357    =/= wne 2447   E.wrex 2545   (/)c0 3456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-v 2791  df-dif 3156  df-nul 3457
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