Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rabeq0OLD Structured version   Visualization version   GIF version

Theorem rabeq0OLD 3936
 Description: Obsolete proof of rabeq0 3933 as of 16-Jul-2021. (Contributed by Jeff Madsen, 7-Jun-2010.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rabeq0OLD ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)

Proof of Theorem rabeq0OLD
StepHypRef Expression
1 ralnex 2986 . 2 (∀𝑥𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥𝐴 𝜑)
2 rabn0OLD 3935 . . 3 ({𝑥𝐴𝜑} ≠ ∅ ↔ ∃𝑥𝐴 𝜑)
32necon1bbii 2839 . 2 (¬ ∃𝑥𝐴 𝜑 ↔ {𝑥𝐴𝜑} = ∅)
41, 3bitr2i 265 1 ({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1480  ∀wral 2907  ∃wrex 2908  {crab 2911  ∅c0 3893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-nul 3894 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator