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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfbi6 | Structured version Visualization version GIF version |
Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
Ref | Expression |
---|---|
bj-dfbi6 | ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-dfbi5 31729 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓))) | |
2 | id 22 | . . . 4 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓))) | |
3 | animorr 505 | . . . 4 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∨ 𝜓)) | |
4 | 2, 3 | impbid1 214 | . . 3 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓))) |
5 | biimp 204 | . . 3 ⊢ (((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓)) → ((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓))) | |
6 | 4, 5 | impbii 198 | . 2 ⊢ (((𝜑 ∨ 𝜓) → (𝜑 ∧ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓))) |
7 | 1, 6 | bitri 263 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∨ 𝜓) ↔ (𝜑 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → 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: (None) |
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