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| Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) | 
| Ref | Expression | 
|---|---|
| eqnetrrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| eqnetrrd.2 | ⊢ (𝜑 → 𝐴 ≠ 𝐶) | 
| Ref | Expression | 
|---|---|
| eqnetrrd | ⊢ (𝜑 → 𝐵 ≠ 𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqnetrrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | 1 | eqcomd 2742 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) | 
| 3 | eqnetrrd.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
| 4 | 2, 3 | eqnetrd 3007 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐶) | 
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