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Mirrors > Home > MPE Home > Th. List > dfbi3 | Structured version Visualization version GIF version |
Description: An alternate definition of the biconditional. Theorem *5.23 of [WhiteheadRussell] p. 124. (Contributed by NM, 27-Jun-2002.) (Proof shortened by Wolf Lammen, 3-Nov-2013.) (Proof shortened by NM, 29-Oct-2021.) |
Ref | Expression |
---|---|
dfbi3 | ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | con34b 319 | . . 3 ⊢ ((𝜓 → 𝜑) ↔ (¬ 𝜑 → ¬ 𝜓)) | |
2 | 1 | anbi2i 626 | . 2 ⊢ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓))) |
3 | dfbi2 478 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) | |
4 | cases2 1048 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 → 𝜓) ∧ (¬ 𝜑 → ¬ 𝜓))) | |
5 | 2, 3, 4 | 3bitr4i 306 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 |
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 210 df-an 400 df-or 848 |
This theorem is referenced by: pm5.24 1051 nanbi 1496 raaan2 4422 2reu4lem 4423 ifbi 4447 sqf11 25975 bj-dfbi4 34440 |
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