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| Mirrors > Home > MPE Home > Th. List > eqeq2d | Structured version Visualization version GIF version | ||
| Description: Deduction from equality to equivalence of equalities. (Contributed by NM, 27-Dec-1993.) Allow shortening of eqeq2 2749. (Revised by Wolf Lammen, 19-Nov-2019.) |
| Ref | Expression |
|---|---|
| eqeq2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| eqeq2d | ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqeq2d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | eqeq1d 2739 | . 2 ⊢ (𝜑 → (𝐴 = 𝐶 ↔ 𝐵 = 𝐶)) |
| 3 | eqcom 2744 | . 2 ⊢ (𝐶 = 𝐴 ↔ 𝐴 = 𝐶) | |
| 4 | eqcom 2744 | . 2 ⊢ (𝐶 = 𝐵 ↔ 𝐵 = 𝐶) | |
| 5 | 2, 3, 4 | 3bitr4g 314 | 1 ⊢ (𝜑 → (𝐶 = 𝐴 ↔ 𝐶 = 𝐵)) |
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