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Metamath Proof Explorer
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012) (Proof shortened by Wolf Lammen, 19-Nov-2019)
|
|
Ref |
Expression |
|
Hypotheses |
3netr3d.1 |
|- ( ph -> A =/= B ) |
|
|
3netr3d.2 |
|- ( ph -> A = C ) |
|
|
3netr3d.3 |
|- ( ph -> B = D ) |
|
Assertion |
3netr3d |
|- ( ph -> C =/= D ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
3netr3d.1 |
|- ( ph -> A =/= B ) |
| 2 |
|
3netr3d.2 |
|- ( ph -> A = C ) |
| 3 |
|
3netr3d.3 |
|- ( ph -> B = D ) |
| 4 |
1 3
|
neeqtrd |
|- ( ph -> A =/= D ) |
| 5 |
2 4
|
eqnetrrd |
|- ( ph -> C =/= D ) |