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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 399 | . 2 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
2 | 1 | albii 1813 | 1 ⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 |
This theorem depends on definitions: df-bi 206 df-an 396 |
This theorem is referenced by: alinexa 1837 raln 3063 rexab 3685 n0el 4356 ballotlem2 34017 |
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