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Metamath Proof Explorer

Theorem rzalALT

Description: Alternate proof of rzal . Shorter, but requiring df-clel , ax-8 . (Contributed by NM, 11-Mar-1997) (Proof shortened by Andrew Salmon, 26-Jun-2011) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	rzalALT	$⊢ A = \emptyset \to \forall x \in A φ$

Proof

Step	Hyp	Ref	Expression
1		ne0i	$⊢ x \in A \to A \neq \emptyset$
2	1	necon2bi	$⊢ A = \emptyset \to \neg x \in A$
3	2	pm2.21d	$⊢ A = \emptyset \to (x \in A \to φ)$
4	3	ralrimiv	$⊢ A = \emptyset \to \forall x \in A φ$