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| Mirrors > Home > MPE Home > Th. List > eqcomd | Structured version Visualization version GIF version | ||
| Description: Deduction from commutative law for class equality. (Contributed by NM, 15-Aug-1994.) Allow shortening of eqcom 2744. (Revised by Wolf Lammen, 19-Nov-2019.) |
| Ref | Expression |
|---|---|
| eqcomd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| eqcomd | ⊢ (𝜑 → 𝐵 = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ 𝐴 = 𝐴 | |
| 2 | eqcomd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 3 | 2 | eqeq1d 2739 | . 2 ⊢ (𝜑 → (𝐴 = 𝐴 ↔ 𝐵 = 𝐴)) |
| 4 | 1, 3 | mpbii 233 | 1 ⊢ (𝜑 → 𝐵 = 𝐴) |
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