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 4451 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 4298 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → 𝐴 ≠ ∅) | |
3 | 2 | necon2bi 3044 | . . 3 ⊢ (𝐴 = ∅ → ¬ 𝑥 ∈ 𝐴) |
4 | 3 | pm2.21d 121 | . 2 ⊢ (𝐴 = ∅ → (𝑥 ∈ 𝐴 → 𝜑)) |
5 | 1, 4 | ralrimi 3214 | 1 ⊢ (𝐴 = ∅ → ∀𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1531 Ⅎwnf 1778 ∈ wcel 2108 ∀wral 3136 ∅c0 4289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-12 2170 ax-ext 2791 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-nf 1779 df-sb 2064 df-clab 2798 df-cleq 2812 df-clel 2891 df-ne 3015 df-ral 3141 df-dif 3937 df-nul 4290 |
This theorem is referenced by: stoweidlem18 42294 stoweidlem28 42304 stoweidlem55 42331 |
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