Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eqnetrrd | GIF version |
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 2086 | . 2 ⊢ (𝜑 → 𝐵 = 𝐴) |
3 | eqnetrrd.2 | . 2 ⊢ (𝜑 → 𝐴 ≠ 𝐶) | |
4 | 2, 3 | eqnetrd 2269 | 1 ⊢ (𝜑 → 𝐵 ≠ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ≠ wne 2245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-5 1376 ax-gen 1378 ax-4 1440 ax-17 1459 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 df-ne 2246 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |