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Theorem rzalALT 4511
Description: Alternate proof of rzal 4510. Shorter, but requiring df-clel 2805, ax-8 2100. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
rzalALT (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rzalALT
StepHypRef Expression
1 ne0i 4336 . . . 4 (𝑥𝐴𝐴 ≠ ∅)
21necon2bi 2967 . . 3 (𝐴 = ∅ → ¬ 𝑥𝐴)
32pm2.21d 121 . 2 (𝐴 = ∅ → (𝑥𝐴𝜑))
43ralrimiv 3141 1 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wral 3057  c0 4324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ne 2937  df-ral 3058  df-dif 3950  df-nul 4325
This theorem is referenced by: (None)
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