![]() |
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 2375 | . 2 ⊢ (𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶) |
4 | 1, 3 | mpbir 146 | 1 ⊢ 𝐴 ≠ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ≠ wne 2360 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-5 1458 ax-gen 1460 ax-4 1521 ax-17 1537 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-cleq 2182 df-ne 2361 |
This theorem is referenced by: eqnetrri 2385 2on0 6445 1n0 6451 basendxnplusgndx 12602 plusgndxnmulrndx 12610 basendxnmulrndx 12611 |
Copyright terms: Public domain | W3C validator |