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Theorem rabeq0 3572
 Description: Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
Assertion
Ref Expression
rabeq0 ({x A φ} = x A ¬ φ)

Proof of Theorem rabeq0
StepHypRef Expression
1 ralnex 2624 . 2 (x A ¬ φ ↔ ¬ x A φ)
2 rabn0 3570 . . 3 ({x A φ} ≠ x A φ)
32necon1bbii 2568 . 2 x A φ ↔ {x A φ} = )
41, 3bitr2i 241 1 ({x A φ} = x A ¬ φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176   = wceq 1642  ∀wral 2614  ∃wrex 2615  {crab 2618  ∅c0 3550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by:  rabnc  3574  nmembers1lem2  6269
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