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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpbi3 | Structured version Visualization version GIF version |
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 14-Apr-2020.) |
Ref | Expression |
---|---|
ifpbi3 | ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | imbi2 351 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → ((¬ 𝜒 → 𝜑) ↔ (¬ 𝜒 → 𝜓))) | |
2 | 1 | anbi2d 630 | . 2 ⊢ ((𝜑 ↔ 𝜓) → (((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜑)) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜓)))) |
3 | dfifp2 1059 | . 2 ⊢ (if-(𝜒, 𝜃, 𝜑) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜑))) | |
4 | dfifp2 1059 | . 2 ⊢ (if-(𝜒, 𝜃, 𝜓) ↔ ((𝜒 → 𝜃) ∧ (¬ 𝜒 → 𝜓))) | |
5 | 2, 3, 4 | 3bitr4g 316 | 1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜒, 𝜃, 𝜑) ↔ if-(𝜒, 𝜃, 𝜓))) |
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: ifpxorcor 39849 ifpnot23c 39857 ifpdfnan 39859 |
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