This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: Deduction for inequality. (Contributed by NM, 25-Oct-1999) (Proof
shortened by Wolf Lammen, 19-Nov-2019)
|
|
Ref |
Expression |
|
Hypothesis |
neeq1d.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
Assertion |
neeq1d |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
neeq1d.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
| 2 |
1
|
eqeq1d |
⊢ ( 𝜑 → ( 𝐴 = 𝐶 ↔ 𝐵 = 𝐶 ) ) |
| 3 |
2
|
necon3bid |
⊢ ( 𝜑 → ( 𝐴 ≠ 𝐶 ↔ 𝐵 ≠ 𝐶 ) ) |