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| Mirrors > Home > MPE Home > Th. List > imnang | Structured version Visualization version GIF version | ||
| Description: Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.) |
| Ref | Expression |
|---|---|
| imnang | ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | imnan 405 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
| 2 | 1 | albii 1852 | 1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 401 ∀wal 1568 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 |
| This proof depends on definitions: df-bi 210 df-an 402 |
| This theorem is used by: alinexa 1876 raln 3090 rexab 3660 n0el 4319 ballotlem2 34944 |
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