![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
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 4335 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 2 | necon2bi 2972 | . . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) |
4 | 3 | pm2.21d 121 | . 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑)) |
5 | 1, 4 | ralrimi 3255 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 ∀wral 3062 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-dif 3952 df-nul 4324 |
This theorem is referenced by: stoweidlem18 44734 stoweidlem28 44744 stoweidlem55 44771 |
Copyright terms: Public domain | W3C validator |