Theorem ral0

Description: Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005) Avoid df-clel , ax-8 . (Revised by GG, 2-Sep-2024)

	Ref	Expression
Assertion	ral0	$$⊢\forall x\in \varnothing \phi$$

Proof

Step	Hyp	Ref	Expression
1		eqid	$$⊢\varnothing =\varnothing$$
2		rzal	$$⊢\varnothing =\varnothing \to \forall x\in \varnothing \phi$$
3	1 2	ax-mp	$$⊢\forall x\in \varnothing \phi$$