Theorem rzalf 44917

Description: A version of rzal 4532 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 4364	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
3	2	necon2bi 2977	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
4	3	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
5	1, 4	ralrimi 3263	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  ∀wral 3067  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-dif 3979  df-nul 4353

This theorem is referenced by:  stoweidlem18  45939  stoweidlem28  45949  stoweidlem55  45976