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Theorem rzalf 38698
Description: A version of rzal 4051 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 3903 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
32necon2bi 2820 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
43pm2.21d 118 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
51, 4ralrimi 2953 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wnf 1705  wcel 1987  wral 2908  c0 3897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-v 3192  df-dif 3563  df-nul 3898
This theorem is referenced by:  stoweidlem18  39572  stoweidlem28  39582  stoweidlem55  39609
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