Theorem rzalALT 4490

Description: Alternate proof of rzal 4489. Shorter, but requiring df-clel 2808, ax-8 2109. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
rzalALT	⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzalALT

Step	Hyp	Ref	Expression
1		ne0i 4321	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
2	1	necon2bi 2961	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
3	2	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
4	3	ralrimiv 3132	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107  ∀wral 3050  ∅c0 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-dif 3934  df-nul 4314

This theorem is referenced by: (None)