| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > eqeq12d | Structured version Visualization version GIF version | ||
| Description: A useful inference for substituting definitions into an equality. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 23-Oct-2024.) |
| Ref | Expression |
|---|---|
| eqeq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| eqeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
| Ref | Expression |
|---|---|
| eqeq12d | ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | eqeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
| 3 | 1, 2 | eqeqan12d 2751 | . 2 ⊢ ((𝜑 ∧ 𝜑) → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| 4 | 3 | anidms 566 | 1 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐷)) |
| Copyright terms: Public domain | W3C validator |