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Mirrors > Home > MPE Home > Th. List > pm5.17 | Structured version Visualization version GIF version |
Description: Theorem *5.17 of [WhiteheadRussell] p. 124. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 3-Jan-2013.) |
Ref | Expression |
---|---|
pm5.17 | ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bicom 211 | . 2 ⊢ ((𝜑 ↔ ¬ 𝜓) ↔ (¬ 𝜓 ↔ 𝜑)) | |
2 | dfbi2 658 | . 2 ⊢ ((¬ 𝜓 ↔ 𝜑) ↔ ((¬ 𝜓 → 𝜑) ∧ (𝜑 → ¬ 𝜓))) | |
3 | orcom 401 | . . . 4 ⊢ ((𝜑 ∨ 𝜓) ↔ (𝜓 ∨ 𝜑)) | |
4 | df-or 384 | . . . 4 ⊢ ((𝜓 ∨ 𝜑) ↔ (¬ 𝜓 → 𝜑)) | |
5 | 3, 4 | bitr2i 264 | . . 3 ⊢ ((¬ 𝜓 → 𝜑) ↔ (𝜑 ∨ 𝜓)) |
6 | imnan 437 | . . 3 ⊢ ((𝜑 → ¬ 𝜓) ↔ ¬ (𝜑 ∧ 𝜓)) | |
7 | 5, 6 | anbi12i 729 | . 2 ⊢ (((¬ 𝜓 → 𝜑) ∧ (𝜑 → ¬ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓))) |
8 | 1, 2, 7 | 3bitrri 286 | 1 ⊢ (((𝜑 ∨ 𝜓) ∧ ¬ (𝜑 ∧ 𝜓)) ↔ (𝜑 ↔ ¬ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ 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: nbi2 932 odd2np1 14903 ordtconlem1 29298 sgnneg 29929 |
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