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Mirrors > Home > ILE Home > Th. List > neeqtrri | GIF version |
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012.) |
Ref | Expression |
---|---|
neeqtrr.1 | ⊢ 𝐴 ≠ 𝐵 |
neeqtrr.2 | ⊢ 𝐶 = 𝐵 |
Ref | Expression |
---|---|
neeqtrri | ⊢ 𝐴 ≠ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neeqtrr.1 | . 2 ⊢ 𝐴 ≠ 𝐵 | |
2 | neeqtrr.2 | . . 3 ⊢ 𝐶 = 𝐵 | |
3 | 2 | eqcomi 2085 | . 2 ⊢ 𝐵 = 𝐶 |
4 | 1, 3 | neeqtri 2272 | 1 ⊢ 𝐴 ≠ 𝐶 |
Colors of variables: wff set class |
Syntax hints: = 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: pnfnemnf 8851 |
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