Metamath Proof Explorer
Description: Substitution of equal classes into the negation of a binary relation.
(Contributed by Glauco Siliprandi, 3-Jan-2021)
|
|
Ref |
Expression |
|
Hypotheses |
eqnbrtrd.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
|
eqnbrtrd.2 |
⊢ ( 𝜑 → ¬ 𝐵 𝑅 𝐶 ) |
|
Assertion |
eqnbrtrd |
⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐶 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
eqnbrtrd.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
2 |
|
eqnbrtrd.2 |
⊢ ( 𝜑 → ¬ 𝐵 𝑅 𝐶 ) |
3 |
1
|
breq1d |
⊢ ( 𝜑 → ( 𝐴 𝑅 𝐶 ↔ 𝐵 𝑅 𝐶 ) ) |
4 |
2 3
|
mtbird |
⊢ ( 𝜑 → ¬ 𝐴 𝑅 𝐶 ) |