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| Mirrors > Home > MPE Home > Th. List > ralnex | Structured version Visualization version GIF version | ||
| Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.) (Proof shortened by BJ, 16-Jul-2021.) |
| Ref | Expression |
|---|---|
| ralnex | ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | raln 3069 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 2 | alnex 1781 | . . 3 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 3 | df-rex 3071 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
| 4 | 2, 3 | xchbinxr 335 | . 2 ⊢ (∀𝑥 ¬ (𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
| 5 | 1, 4 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜑) |
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