Theorem rzalf 41281

Description: A version of rzal 4455 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypothesis

Ref	Expression
rzalf.1	⊢ Ⅎ𝑥 𝐴 = ∅

Assertion

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

Proof of Theorem rzalf

Step	Hyp	Ref	Expression
1		rzalf.1	. 2 ⊢ Ⅎ𝑥 𝐴 = ∅
2		ne0i 4302	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
3	2	necon2bi 3048	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
4	3	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
5	1, 4	ralrimi 3218	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1537  Ⅎwnf 1784   ∈ wcel 2114  ∀wral 3140  ∅c0 4293

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-ne 3019  df-ral 3145  df-dif 3941  df-nul 4294

This theorem is referenced by:  stoweidlem18  42310  stoweidlem28  42320  stoweidlem55  42347