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Mirrors > Home > MPE Home > Th. List > Mathboxes > andiff | Structured version Visualization version GIF version |
Description: Adding biconditional when antecedents are conjuncted. (Contributed by metakunt, 16-Apr-2024.) |
Ref | Expression |
---|---|
andiff.1 | ⊢ (𝜑 → (𝜒 → 𝜃)) |
andiff.2 | ⊢ (𝜓 → (𝜃 → 𝜒)) |
Ref | Expression |
---|---|
andiff | ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | andiff.1 | . . 3 ⊢ (𝜑 → (𝜒 → 𝜃)) | |
2 | andiff.2 | . . 3 ⊢ (𝜓 → (𝜃 → 𝜒)) | |
3 | 1, 2 | anim12i 616 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝜒 → 𝜃) ∧ (𝜃 → 𝜒))) |
4 | dfbi2 478 | . 2 ⊢ ((𝜒 ↔ 𝜃) ↔ ((𝜒 → 𝜃) ∧ (𝜃 → 𝜒))) | |
5 | 3, 4 | sylibr 237 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝜒 ↔ 𝜃)) |
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: (None) |
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