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Theorem rzalf 39675
Description: A version of rzal 4217 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 4064 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
32necon2bi 2962 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
43pm2.21d 118 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
51, 4ralrimi 3095 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1632  wnf 1857  wcel 2139  wral 3050  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-v 3342  df-dif 3718  df-nul 4059
This theorem is referenced by:  stoweidlem18  40738  stoweidlem28  40748  stoweidlem55  40775
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