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Mirrors > Home > ILE Home > Th. List > 3anbi12d | GIF version |
Description: Deduction conjoining and adding a conjunct to equivalences. (Contributed by NM, 8-Sep-2006.) |
Ref | Expression |
---|---|
3anbi12d.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
3anbi12d.2 | ⊢ (𝜑 → (𝜃 ↔ 𝜏)) |
Ref | Expression |
---|---|
3anbi12d | ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3anbi12d.1 | . 2 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 3anbi12d.2 | . 2 ⊢ (𝜑 → (𝜃 ↔ 𝜏)) | |
3 | biidd 172 | . 2 ⊢ (𝜑 → (𝜂 ↔ 𝜂)) | |
4 | 1, 2, 3 | 3anbi123d 1312 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃 ∧ 𝜂) ↔ (𝜒 ∧ 𝜏 ∧ 𝜂))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∧ w3a 978 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 |
This theorem depends on definitions: df-bi 117 df-3an 980 |
This theorem is referenced by: 3anbi1d 1316 3anbi2d 1317 fseq1m1p1 10078 unitsubm 13110 |
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