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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 402 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | 1 | albii 1819 | 1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 |
This theorem depends on definitions: df-bi 209 df-an 399 |
This theorem is referenced by: alinexa 1842 raln 3158 n0el 4324 ballotlem2 31750 wl-dfrexsb 34855 |
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