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Mirrors > Home > MPE Home > Th. List > anbi1d | Structured version Visualization version GIF version |
Description: Deduction adding a right 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 |
---|---|
anbi1d | ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜃))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anbid.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | a1d 25 | . 2 ⊢ (𝜑 → (𝜃 → (𝜓 ↔ 𝜒))) |
3 | 2 | pm5.32rd 581 | 1 ⊢ (𝜑 → ((𝜓 ∧ 𝜃) ↔ (𝜒 ∧ 𝜃))) |
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