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Theorem rabeq0OLD 26351
Description: Condition for a restricted class abstraction to be empty. (Moved to rabeq0 3476 in main set.mm and may be deleted by mathbox owner, JM. --NM 1-Apr-2013.) (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rabeq0OLD  |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem rabeq0OLD
StepHypRef Expression
1 rabeq0 3476 1  |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1623   A.wral 2543   {crab 2547   (/)c0 3455
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 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-nul 3456
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