Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rzalf | Structured version Visualization version GIF version |
Description: A version of rzal 4025 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
rzalf.1 | ⊢ Ⅎ𝑥 𝐴 = ∅ |
Ref | Expression |
---|---|
rzalf | ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rzalf.1 | . 2 ⊢ Ⅎ𝑥 𝐴 = ∅ | |
2 | ne0i 3880 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 2 | necon2bi 2812 | . . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) |
4 | 3 | pm2.21d 117 | . 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑)) |
5 | 1, 4 | ralrimi 2940 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 Ⅎwnf 1699 ∈ wcel 1977 ∀wral 2896 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-v 3175 df-dif 3543 df-nul 3875 |
This theorem is referenced by: stoweidlem18 38911 stoweidlem28 38921 stoweidlem55 38948 |
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