Theorem bj-abf 34218

Description: Shorter proof of abf 4354 (which should be kept as abfALT). (Contributed by BJ, 24-Jul-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bj-abf.1	⊢ ¬ 𝜑

Assertion

Ref	Expression
bj-abf	⊢ {𝑥 ∣ 𝜑} = ∅

Proof of Theorem bj-abf

Step	Hyp	Ref	Expression
1		bj-ab0 34217	. 2 ⊢ (∀𝑥 ¬ 𝜑 → {𝑥 ∣ 𝜑} = ∅)
2		bj-abf.1	. 2 ⊢ ¬ 𝜑
3	1, 2	mpg 1792	1 ⊢ {𝑥 ∣ 𝜑} = ∅

Syntax hints:  ¬ wn 3   = wceq 1531  {cab 2797  ∅c0 4289

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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-dif 3937  df-nul 4290

This theorem is referenced by: (None)