Theorem rzalf

Description: A version of rzal using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017)

	Ref	Expression
Hypothesis	rzalf.1	$$⊢ℲxA=\varnothing$$
Assertion	rzalf	$$⊢A=\varnothing \to \forall x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		rzalf.1	$$⊢ℲxA=\varnothing$$
2		ne0i	$$⊢x\in A\to A\ne \varnothing$$
3	2	necon2bi	$$⊢A=\varnothing \to \neg x\in A$$
4	3	pm2.21d	$$⊢A=\varnothing \to \left(x\in A\to \phi \right)$$
5	1 4	ralrimi	$$⊢A=\varnothing \to \forall x\in A\phi$$