Metamath Proof Explorer
Description: Substitution of equal classes into an inequality. (Contributed by NM, 4-Jul-2012)
|
|
Ref |
Expression |
|
Hypotheses |
neeqtrrd.1 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
|
|
neeqtrrd.2 |
⊢ ( 𝜑 → 𝐶 = 𝐵 ) |
|
Assertion |
neeqtrrd |
⊢ ( 𝜑 → 𝐴 ≠ 𝐶 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
neeqtrrd.1 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
2 |
|
neeqtrrd.2 |
⊢ ( 𝜑 → 𝐶 = 𝐵 ) |
3 |
2
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = 𝐶 ) |
4 |
1 3
|
neeqtrd |
⊢ ( 𝜑 → 𝐴 ≠ 𝐶 ) |