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| Mirrors > Home > MPE Home > Th. List > anbi2d | Structured version Visualization version GIF version | ||
| Description: Deduction adding a left conjunct to both sides of a logical equivalence. (Contributed by NM, 11-May-1993.) (Proof shortened by Wolf Lammen, 16-Nov-2013.) |
| Ref | Expression |
|---|---|
| anbid.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| anbi2d | ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | anbid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
| 2 | 1 | a1d 25 | . 2 ⊢ (𝜑 → (𝜃 → (𝜓 ↔ 𝜒))) |
| 3 | 2 | pm5.32d 577 | 1 ⊢ (𝜑 → ((𝜃 ∧ 𝜓) ↔ (𝜃 ∧ 𝜒))) |
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