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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-dfbi4 | Structured version Visualization version GIF version |
Description: Alternate definition of the biconditional. (Contributed by BJ, 4-Oct-2019.) |
Ref | Expression |
---|---|
bj-dfbi4 | ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfbi3 1050 | . 2 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓))) | |
2 | pm4.56 989 | . . 3 ⊢ ((¬ 𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 ∨ 𝜓)) | |
3 | 2 | orbi2i 913 | . 2 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) |
4 | 1, 3 | bitri 278 | 1 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ ¬ (𝜑 ∨ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ 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: bj-dfbi5 34518 |
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