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 = ∅ → ∀ 𝑥 ∈ 𝐴 𝜑 )

Proof

Step	Hyp	Ref	Expression
1		ne0i	⊢ ( 𝑥 ∈ 𝐴 → 𝐴 ≠ ∅ )
2	1	necon2bi	⊢ ( 𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴 )
3	2	pm2.21d	⊢ ( 𝐴 = ∅ → ( 𝑥 ∈ 𝐴 → 𝜑 ) )
4	3	ralrimiv	⊢ ( 𝐴 = ∅ → ∀ 𝑥 ∈ 𝐴 𝜑 )