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Mirrors > Home > MPE Home > Th. List > anor | Structured version Visualization version GIF version |
Description: Conjunction in terms of disjunction (De Morgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 3-Nov-2012.) |
Ref | Expression |
---|---|
anor | ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ianor 508 | . . 3 ⊢ (¬ (𝜑 ∧ 𝜓) ↔ (¬ 𝜑 ∨ ¬ 𝜓)) | |
2 | 1 | bicomi 213 | . 2 ⊢ ((¬ 𝜑 ∨ ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) |
3 | 2 | con2bii 346 | 1 ⊢ ((𝜑 ∧ 𝜓) ↔ ¬ (¬ 𝜑 ∨ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 ∧ wa 383 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 |
This theorem is referenced by: pm3.1 518 pm3.11 519 dn1 1000 3anor 1047 bropopvvv 7142 2wlkonot3v 26402 2spthonot3v 26403 ifpananb 36870 iunrelexp0 37013 |
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