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Mirrors > Home > MPE Home > Th. List > pm5.75 | Structured version Visualization version GIF version |
Description: Theorem *5.75 of [WhiteheadRussell] p. 126. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 23-Dec-2012.) (Proof shortened by Kyle Wyonch, 12-Feb-2021.) |
Ref | Expression |
---|---|
pm5.75 | ⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anbi1 632 | . . 3 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ ((𝜓 ∨ 𝜒) ∧ ¬ 𝜓))) | |
2 | biorf 934 | . . . . 5 ⊢ (¬ 𝜓 → (𝜒 ↔ (𝜓 ∨ 𝜒))) | |
3 | 2 | bicomd 222 | . . . 4 ⊢ (¬ 𝜓 → ((𝜓 ∨ 𝜒) ↔ 𝜒)) |
4 | 3 | pm5.32ri 576 | . . 3 ⊢ (((𝜓 ∨ 𝜒) ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓)) |
5 | 1, 4 | bitrdi 287 | . 2 ⊢ ((𝜑 ↔ (𝜓 ∨ 𝜒)) → ((𝜑 ∧ ¬ 𝜓) ↔ (𝜒 ∧ ¬ 𝜓))) |
6 | abai 824 | . . 3 ⊢ ((𝜒 ∧ ¬ 𝜓) ↔ (𝜒 ∧ (𝜒 → ¬ 𝜓))) | |
7 | 6 | rbaib 539 | . 2 ⊢ ((𝜒 → ¬ 𝜓) → ((𝜒 ∧ ¬ 𝜓) ↔ 𝜒)) |
8 | 5, 7 | sylan9bbr 511 | 1 ⊢ (((𝜒 → ¬ 𝜓) ∧ (𝜑 ↔ (𝜓 ∨ 𝜒))) → ((𝜑 ∧ ¬ 𝜓) ↔ 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 |
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 206 df-an 397 df-or 845 |
This theorem is referenced by: (None) |
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