Theorem abfOLD 4293

Description: Obsolete version of abf 4292 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 4266	. 2 ⊢ ({𝑥 ∣ 𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
2		abfOLD.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpgbir 1802	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1539  {cab 2736  ∅c0 4221

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-dif 3857  df-nul 4222

This theorem is referenced by: (None)