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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 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑)) | |
2 | imnang 1804 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → ¬ 𝜑) ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
3 | 1, 2 | bitri 267 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 387 ∀wal 1505 ∈ wcel 2050 ∀wral 3088 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ral 3093 |
This theorem is referenced by: ralnex 3183 rabeq0 4224 |
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