Theorem rzalf 42020

Description: A version of rzal 4402 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 4234	. . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅)
3	2	necon2bi 2982	. . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴)
4	3	pm2.21d 121	. 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑))
5	1, 4	ralrimi 3145	1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:   → wi 4   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112  ∀wral 3071  ∅c0 4226

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-12 2176  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-ne 2953  df-ral 3076  df-dif 3862  df-nul 4227

This theorem is referenced by:  stoweidlem18  43027  stoweidlem28  43037  stoweidlem55  43064