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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > biadanid | Structured version Visualization version GIF version |
Description: Deduction associated with biadani 820. Add a conjunction to an equivalence. (Contributed by Thierry Arnoux, 16-Jun-2024.) |
Ref | Expression |
---|---|
biadanid.1 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
biadanid.2 | ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) |
Ref | Expression |
---|---|
biadanid | ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biadanid.1 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
2 | biadanid.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝜒) → (𝜓 ↔ 𝜃)) | |
3 | 2 | biimpa 481 | . . . . 5 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜓) → 𝜃) |
4 | 3 | an32s 652 | . . . 4 ⊢ (((𝜑 ∧ 𝜓) ∧ 𝜒) → 𝜃) |
5 | 1, 4 | mpdan 687 | . . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜃) |
6 | 1, 5 | jca 516 | . 2 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ∧ 𝜃)) |
7 | 2 | biimpar 482 | . . 3 ⊢ (((𝜑 ∧ 𝜒) ∧ 𝜃) → 𝜓) |
8 | 7 | anasss 471 | . 2 ⊢ ((𝜑 ∧ (𝜒 ∧ 𝜃)) → 𝜓) |
9 | 6, 8 | impbida 801 | 1 ⊢ (𝜑 → (𝜓 ↔ (𝜒 ∧ 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 |
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 401 |
This theorem is referenced by: ist0cld 31289 |
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