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Theorem rzalf 45022
Description: A version of rzal 4509 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 4341 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
32necon2bi 2971 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
43pm2.21d 121 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
51, 4ralrimi 3257 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wnf 1783  wcel 2108  wral 3061  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-dif 3954  df-nul 4334
This theorem is referenced by:  stoweidlem18  46033  stoweidlem28  46043  stoweidlem55  46070
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