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Theorem rabeq0OLD 25675
Description: Condition for a restricted class abstraction to be empty. (Moved to rabeq0 3552 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 3552 1  |-  ( { x  e.  A  |  ph }  =  (/)  <->  A. x  e.  A  -.  ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    = wceq 1642   A.wral 2619   {crab 2623   (/)c0 3531
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-dif 3231  df-nul 3532
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