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Mirrors > Home > MPE Home > Th. List > ifpbi123d | Structured version Visualization version GIF version |
Description: Equality deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) |
Ref | Expression |
---|---|
ifpbi123d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜏)) |
ifpbi123d.2 | ⊢ (𝜑 → (𝜒 ↔ 𝜂)) |
ifpbi123d.3 | ⊢ (𝜑 → (𝜃 ↔ 𝜁)) |
Ref | Expression |
---|---|
ifpbi123d | ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ifpbi123d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜏)) | |
2 | ifpbi123d.2 | . . . 4 ⊢ (𝜑 → (𝜒 ↔ 𝜂)) | |
3 | 1, 2 | anbi12d 743 | . . 3 ⊢ (𝜑 → ((𝜓 ∧ 𝜒) ↔ (𝜏 ∧ 𝜂))) |
4 | 1 | notbid 307 | . . . 4 ⊢ (𝜑 → (¬ 𝜓 ↔ ¬ 𝜏)) |
5 | ifpbi123d.3 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁)) | |
6 | 4, 5 | anbi12d 743 | . . 3 ⊢ (𝜑 → ((¬ 𝜓 ∧ 𝜃) ↔ (¬ 𝜏 ∧ 𝜁))) |
7 | 3, 6 | orbi12d 742 | . 2 ⊢ (𝜑 → (((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃)) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁)))) |
8 | df-ifp 1007 | . 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 ∧ 𝜒) ∨ (¬ 𝜓 ∧ 𝜃))) | |
9 | df-ifp 1007 | . 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 ∧ 𝜂) ∨ (¬ 𝜏 ∧ 𝜁))) | |
10 | 7, 8, 9 | 3bitr4g 302 | 1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 if-wif 1006 |
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 196 df-or 384 df-an 385 df-ifp 1007 |
This theorem is referenced by: 1wlkslem1 40809 1wlkslem2 40810 1wlksfval 40811 is1wlk 40813 1wlkres 40879 red1wlk 40881 1wlkp1lem8 40889 crctcsh1wlkn0lem4 41016 crctcsh1wlkn0lem5 41017 crctcsh1wlkn0lem6 41018 11wlkdlem4 41307 |
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