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Mirrors > Home > MPE Home > Th. List > raln | Structured version Visualization version GIF version |
Description: Restricted universally quantified negation expressed as a universally quantified negation. (Contributed by BJ, 16-Jul-2021.) |
Ref | Expression |
---|---|
raln | ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3057 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | imnang 1837 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | 1, 2 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1532 ∈ wcel 2099 ∀wral 3056 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ral 3057 |
This theorem is referenced by: ralnex 3067 rabeq0w 4379 rabeq0 4380 |
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