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Mirrors > Home > MPE Home > Th. List > ifpbi123d | Structured version Visualization version GIF version |
Description: Equivalence deduction for conditional operator for propositions. (Contributed by AV, 30-Dec-2020.) (Proof shortened by Wolf Lammen, 17-Apr-2024.) |
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 | imbi12d 345 | . . 3 ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜏 → 𝜂))) |
4 | ifpbi123d.3 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜁)) | |
5 | 1, 4 | orbi12d 916 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜏 ∨ 𝜁))) |
6 | 3, 5 | anbi12d 631 | . 2 ⊢ (𝜑 → (((𝜓 → 𝜒) ∧ (𝜓 ∨ 𝜃)) ↔ ((𝜏 → 𝜂) ∧ (𝜏 ∨ 𝜁)))) |
7 | dfifp3 1063 | . 2 ⊢ (if-(𝜓, 𝜒, 𝜃) ↔ ((𝜓 → 𝜒) ∧ (𝜓 ∨ 𝜃))) | |
8 | dfifp3 1063 | . 2 ⊢ (if-(𝜏, 𝜂, 𝜁) ↔ ((𝜏 → 𝜂) ∧ (𝜏 ∨ 𝜁))) | |
9 | 6, 7, 8 | 3bitr4g 314 | 1 ⊢ (𝜑 → (if-(𝜓, 𝜒, 𝜃) ↔ if-(𝜏, 𝜂, 𝜁))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 844 if-wif 1060 |
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 397 df-or 845 df-ifp 1061 |
This theorem is referenced by: ifpbi23d 1079 wkslem1 27974 wkslem2 27975 iswlk 27977 wlkres 28038 redwlk 28040 wlkp1lem8 28048 crctcshwlkn0lem4 28178 crctcshwlkn0lem5 28179 crctcshwlkn0lem6 28180 1wlkdlem4 28504 pfxwlk 33085 satfv1fvfmla1 33385 ifpbi123 41097 |
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