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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpbi1 | Structured version Visualization version GIF version |
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
Ref | Expression |
---|---|
ifpbi1 | ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbi1 350 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → ((𝜑 → 𝜒) ↔ (𝜓 → 𝜒))) | |
2 | notbi 321 | . . . . 5 ⊢ ((𝜑 ↔ 𝜓) ↔ (¬ 𝜑 ↔ ¬ 𝜓)) | |
3 | 2 | biimpi 218 | . . . 4 ⊢ ((𝜑 ↔ 𝜓) → (¬ 𝜑 ↔ ¬ 𝜓)) |
4 | 3 | imbi1d 344 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → ((¬ 𝜑 → 𝜃) ↔ (¬ 𝜓 → 𝜃))) |
5 | 1, 4 | anbi12d 632 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜃)) ↔ ((𝜓 → 𝜒) ∧ (¬ 𝜓 → 𝜃)))) |
6 | dfifp2 1059 | . 2 ⊢ (if-(𝜑, 𝜒, 𝜃) ↔ ((𝜑 → 𝜒) ∧ (¬ 𝜑 → 𝜃))) | |
7 | dfifp2 1059 | . 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 → 𝜒) ∧ (¬ 𝜓 → 𝜃))) | |
8 | 5, 6, 7 | 3bitr4g 316 | 1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) ↔ if-(𝜓, 𝜒, 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 if-wif 1057 |
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 209 df-an 399 df-or 844 df-ifp 1058 |
This theorem is referenced by: ifpimim 39881 axfrege52a 40208 |
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