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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.) |
Ref | Expression |
---|---|
dfbi3 | ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xor 931 | . 2 ⊢ (¬ (𝜑 ↔ ¬ 𝜓) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))) | |
2 | pm5.18 370 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ¬ (𝜑 ↔ ¬ 𝜓)) | |
3 | notnotb 303 | . . . 4 ⊢ (𝜓 ↔ ¬ ¬ 𝜓) | |
4 | 3 | anbi2i 726 | . . 3 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ¬ ¬ 𝜓)) |
5 | ancom 465 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ (¬ 𝜓 ∧ ¬ 𝜑)) | |
6 | 4, 5 | orbi12i 542 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ ¬ ¬ 𝜓) ∨ (¬ 𝜓 ∧ ¬ 𝜑))) |
7 | 1, 2, 6 | 3bitr4i 291 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ 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: pm5.24 934 4exmid 977 nanbi 1446 ifbi 4057 sqf11 24665 bj-dfbi4 31728 raaan2 39824 2reu4a 39838 |
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