![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpbi13 | Structured version Visualization version GIF version |
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
Ref | Expression |
---|---|
ifpbi13 | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 484 | . . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓)) | |
2 | 1 | imbi1d 342 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜑 → 𝜏) ↔ (𝜓 → 𝜏))) |
3 | notbi 319 | . . . . 5 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) | |
4 | imbi12 347 | . . . . 5 ⊢ ((¬ 𝜑 ↔ ¬ 𝜓) → ((𝜒 ↔ 𝜃) → ((¬ 𝜑 → 𝜒) ↔ (¬ 𝜓 → 𝜃)))) | |
5 | 3, 4 | sylbi 216 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → ((𝜒 ↔ 𝜃) → ((¬ 𝜑 → 𝜒) ↔ (¬ 𝜓 → 𝜃)))) |
6 | 5 | imp 408 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((¬ 𝜑 → 𝜒) ↔ (¬ 𝜓 → 𝜃))) |
7 | 2, 6 | anbi12d 632 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (((𝜑 → 𝜏) ∧ (¬ 𝜑 → 𝜒)) ↔ ((𝜓 → 𝜏) ∧ (¬ 𝜓 → 𝜃)))) |
8 | dfifp2 1064 | . 2 ⊢ (if-(𝜑, 𝜏, 𝜒) ↔ ((𝜑 → 𝜏) ∧ (¬ 𝜑 → 𝜒))) | |
9 | dfifp2 1064 | . 2 ⊢ (if-(𝜓, 𝜏, 𝜃) ↔ ((𝜓 → 𝜏) ∧ (¬ 𝜓 → 𝜃))) | |
10 | 7, 8, 9 | 3bitr4g 314 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜑, 𝜏, 𝜒) ↔ if-(𝜓, 𝜏, 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 if-wif 1062 |
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 206 df-an 398 df-or 847 df-ifp 1063 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |