![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > eqnetri | GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
eqnetr.1 | ⊢ 𝐴 = 𝐵 |
eqnetr.2 | ⊢ 𝐵 ≠ 𝐶 |
Ref | Expression |
---|---|
eqnetri | ⊢ 𝐴 ≠ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqnetr.2 | . 2 ⊢ 𝐵 ≠ 𝐶 | |
2 | eqnetr.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
3 | 2 | neeq1i 2295 | . 2 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) |
4 | 1, 3 | mpbir 145 | 1 ⊢ 𝐴 ≠ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1312 ≠ wne 2280 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 586 ax-in2 587 ax-5 1404 ax-gen 1406 ax-4 1468 ax-17 1487 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-cleq 2106 df-ne 2281 |
This theorem is referenced by: eqnetrri 2305 2on0 6275 1n0 6281 basendxnplusgndx 11902 plusgndxnmulrndx 11909 basendxnmulrndx 11910 |
Copyright terms: Public domain | W3C validator |