Theorem rzalf 45602

Description: A version of rzal 4450 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 4295	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
3	2	necon2bi 2989	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
4	3	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
5	1, 4	ralrimi 3262	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1562  Ⅎwnf 1805   ∈ wcel 2144  ∀wral 3078  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-12 2214  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-dif 3909  df-nul 4288

This theorem is referenced by:  stoweidlem18  46597  stoweidlem28  46607  stoweidlem55  46634