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Mirrors > Home > MPE Home > Th. List > Mathboxes > ifpbi23 | Structured version Visualization version GIF version |
Description: Equivalence theorem for conditional logical operators. (Contributed by RP, 15-Apr-2020.) |
Ref | Expression |
---|---|
ifpbi23 | ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜑 ↔ 𝜓)) | |
2 | 1 | imbi2d 342 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((𝜏 → 𝜑) ↔ (𝜏 → 𝜓))) |
3 | simpr 485 | . . . 4 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (𝜒 ↔ 𝜃)) | |
4 | 3 | imbi2d 342 | . . 3 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → ((¬ 𝜏 → 𝜒) ↔ (¬ 𝜏 → 𝜃))) |
5 | 2, 4 | anbi12d 630 | . 2 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (((𝜏 → 𝜑) ∧ (¬ 𝜏 → 𝜒)) ↔ ((𝜏 → 𝜓) ∧ (¬ 𝜏 → 𝜃)))) |
6 | dfifp2 1057 | . 2 ⊢ (if-(𝜏, 𝜑, 𝜒) ↔ ((𝜏 → 𝜑) ∧ (¬ 𝜏 → 𝜒))) | |
7 | dfifp2 1057 | . 2 ⊢ (if-(𝜏, 𝜓, 𝜃) ↔ ((𝜏 → 𝜓) ∧ (¬ 𝜏 → 𝜃))) | |
8 | 5, 6, 7 | 3bitr4g 315 | 1 ⊢ (((𝜑 ↔ 𝜓) ∧ (𝜒 ↔ 𝜃)) → (if-(𝜏, 𝜑, 𝜒) ↔ if-(𝜏, 𝜓, 𝜃))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 if-wif 1055 |
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 208 df-an 397 df-or 843 df-ifp 1056 |
This theorem is referenced by: ifpdfbi 39350 ifpnot23d 39362 ifpdfxor 39364 ifpananb 39383 ifpnannanb 39384 ifpxorxorb 39388 |
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