Theorem ral0OLD 4444

Description: Obsolete version of ral0 4440 as of 2-Sep-2024. (Contributed by NM, 20-Oct-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ral0OLD	⊢ ∀𝑥 ∈ ∅ 𝜑

Proof of Theorem ral0OLD

Step	Hyp	Ref	Expression
1		noel 4261	. . 3 ⊢ ¬ 𝑥 ∈ ∅
2	1	pm2.21i 119	. 2 ⊢ (𝑥 ∈ ∅ → 𝜑)
3	2	rgen 3073	1 ⊢ ∀𝑥 ∈ ∅ 𝜑

Syntax hints:   ∈ wcel 2108  ∀wral 3063  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-dif 3886  df-nul 4254

This theorem is referenced by: (None)