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Theorem rzalf 38199
Description: A version of rzal 4025 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 3880 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
32necon2bi 2812 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
43pm2.21d 117 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
51, 4ralrimi 2940 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wnf 1699  wcel 1977  wral 2896  c0 3874
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-v 3175  df-dif 3543  df-nul 3875
This theorem is referenced by:  stoweidlem18  38911  stoweidlem28  38921  stoweidlem55  38948
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