Theorem rzal

Description: Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	rzal	$$⊢A=\varnothing \to \forall x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		ne0i	$$⊢x\in A\to A\ne \varnothing$$
2	1	necon2bi	$$⊢A=\varnothing \to \neg x\in A$$
3	2	pm2.21d	$$⊢A=\varnothing \to \left(x\in A\to \phi \right)$$
4	3	ralrimiv	$$⊢A=\varnothing \to \forall x\in A\phi$$