![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > pm5.55 | Structured version Visualization version GIF version |
Description: Theorem *5.55 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 20-Jan-2013.) |
Ref | Expression |
---|---|
pm5.55 | ⊢ (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biort 936 | . . . . 5 ⊢ (𝜑 → (𝜑 ↔ (𝜑 ∨ 𝜓))) | |
2 | 1 | bicomd 212 | . . . 4 ⊢ (𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
3 | biorf 419 | . . . . 5 ⊢ (¬ 𝜑 → (𝜓 ↔ (𝜑 ∨ 𝜓))) | |
4 | 3 | bicomd 212 | . . . 4 ⊢ (¬ 𝜑 → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
5 | 2, 4 | nsyl4 155 | . . 3 ⊢ (¬ ((𝜑 ∨ 𝜓) ↔ 𝜓) → ((𝜑 ∨ 𝜓) ↔ 𝜑)) |
6 | 5 | con1i 143 | . 2 ⊢ (¬ ((𝜑 ∨ 𝜓) ↔ 𝜑) → ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
7 | 6 | orri 390 | 1 ⊢ (((𝜑 ∨ 𝜓) ↔ 𝜑) ∨ ((𝜑 ∨ 𝜓) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∨ wo 382 |
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 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |