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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 𝑥 𝐴 = ∅
Assertion
Ref Expression
rzalf (𝐴 = ∅ → ∀𝑥𝐴 𝜑)

Proof of Theorem rzalf
StepHypRef Expression
1 rzalf.1 . 2 𝑥 𝐴 = ∅
2 ne0i 4234 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
32necon2bi 2982 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
43pm2.21d 121 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
51, 4ralrimi 3145 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
 Colors of variables: wff setvar class 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
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