Metamath Proof Explorer
Description: Substitution of equality into both sides of an inequality. (Contributed by NM, 24-Jul-2012)
|
|
Ref |
Expression |
|
Hypotheses |
3netr3g.1 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
|
|
3netr3g.2 |
⊢ 𝐴 = 𝐶 |
|
|
3netr3g.3 |
⊢ 𝐵 = 𝐷 |
|
Assertion |
3netr3g |
⊢ ( 𝜑 → 𝐶 ≠ 𝐷 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
3netr3g.1 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
2 |
|
3netr3g.2 |
⊢ 𝐴 = 𝐶 |
3 |
|
3netr3g.3 |
⊢ 𝐵 = 𝐷 |
4 |
2 3
|
neeq12i |
⊢ ( 𝐴 ≠ 𝐵 ↔ 𝐶 ≠ 𝐷 ) |
5 |
1 4
|
sylib |
⊢ ( 𝜑 → 𝐶 ≠ 𝐷 ) |