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| Mirrors > Home > MPE Home > Th. List > Mathboxes > rzalf | Structured version Visualization version GIF version | ||
| Description: A version of rzal 4509 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 4341 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
| 3 | 2 | necon2bi 2971 | . . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) |
| 4 | 3 | pm2.21d 121 | . 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑)) |
| 5 | 1, 4 | ralrimi 3257 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-dif 3954 df-nul 4334 |
| This theorem is referenced by: stoweidlem18 46033 stoweidlem28 46043 stoweidlem55 46070 |
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