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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-animbi | Structured version Visualization version GIF version |
Description: Conjunction in terms of implication and biconditional. Note that the proof is intuitionistic (use of ax-3 8 comes from the unusual definition of the biconditional in set.mm). (Contributed by BJ, 23-Sep-2023.) |
Ref | Expression |
---|---|
bj-animbi | ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 486 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | pm3.4 810 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 → 𝜓)) | |
3 | 1, 2 | 2thd 268 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ↔ (𝜑 → 𝜓))) |
4 | biimp 218 | . . . . 5 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → (𝜑 → (𝜑 → 𝜓))) | |
5 | 4 | pm2.43d 53 | . . . 4 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → (𝜑 → 𝜓)) |
6 | biimpr 223 | . . . 4 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → ((𝜑 → 𝜓) → 𝜑)) | |
7 | 5, 6 | mpd 15 | . . 3 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → 𝜑) |
8 | 7, 5 | jcai 520 | . 2 ⊢ ((𝜑 ↔ (𝜑 → 𝜓)) → (𝜑 ∧ 𝜓)) |
9 | 3, 8 | impbii 212 | 1 ⊢ ((𝜑 ∧ 𝜓) ↔ (𝜑 ↔ (𝜑 → 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 |
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 |
This theorem is referenced by: bj-currypara 34477 |
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