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Mirrors > Home > ILE Home > Th. List > 3orbi123d | GIF version |
Description: Deduction joining 3 equivalences to form equivalence of disjunctions. (Contributed by NM, 20-Apr-1994.) |
Ref | Expression |
---|---|
bi3d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
bi3d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
bi3d.3 | ⊢ (𝜑 → (𝜂 ↔ 𝜁)) |
Ref | Expression |
---|---|
3orbi123d | ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi3d.1 | . . . 4 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | bi3d.2 | . . . 4 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
3 | 1, 2 | orbi12d 793 | . . 3 ⊢ (𝜑 → ((𝜓 ∨ 𝜃) ↔ (𝜒 ∨ 𝜏))) |
4 | bi3d.3 | . . 3 ⊢ (𝜑 → (𝜂 ↔ 𝜁)) | |
5 | 3, 4 | orbi12d 793 | . 2 ⊢ (𝜑 → (((𝜓 ∨ 𝜃) ∨ 𝜂) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁))) |
6 | df-3or 979 | . 2 ⊢ ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ ((𝜓 ∨ 𝜃) ∨ 𝜂)) | |
7 | df-3or 979 | . 2 ⊢ ((𝜒 ∨ 𝜏 ∨ 𝜁) ↔ ((𝜒 ∨ 𝜏) ∨ 𝜁)) | |
8 | 5, 6, 7 | 3bitr4g 223 | 1 ⊢ (𝜑 → ((𝜓 ∨ 𝜃 ∨ 𝜂) ↔ (𝜒 ∨ 𝜏 ∨ 𝜁))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∨ wo 708 ∨ w3o 977 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 |
This theorem depends on definitions: df-bi 117 df-3or 979 |
This theorem is referenced by: ordtriexmid 4522 ontriexmidim 4523 wetriext 4578 nntri3or 6496 tridc 6901 exmidontriimlem3 7224 exmidontriimlem4 7225 exmidontriim 7226 onntri35 7238 ltsopi 7321 pitri3or 7323 nqtri3or 7397 elz 9257 ztri3or 9298 qtri3or 10245 trilpo 14830 trirec0 14831 reap0 14845 |
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