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Theorem rzalf 43701
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 4335 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
32necon2bi 2972 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
43pm2.21d 121 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
51, 4ralrimi 3255 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wnf 1786  wcel 2107  wral 3062  c0 4323
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-dif 3952  df-nul 4324
This theorem is referenced by:  stoweidlem18  44734  stoweidlem28  44744  stoweidlem55  44771
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