Theorem abfOLD 4404

Description: Obsolete version of abf 4403 as of 28-Jun-2024. (Contributed by NM, 20-Jan-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
abfOLD.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
abfOLD	⊢ {𝑥 ∣ 𝜑} = ∅

Proof of Theorem abfOLD

Step	Hyp	Ref	Expression
1		ab0 4375	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2		abfOLD.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbir 1794	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1534  {cab 2705  ∅c0 4323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-dif 3950  df-nul 4324

This theorem is referenced by: (None)