Metamath Proof Explorer
Description: A transitivity relation for equivalences. (Contributed by Mario
Carneiro, 9-Jul-2014)
|
|
Ref |
Expression |
|
Hypotheses |
ersymb.1 |
⊢ ( 𝜑 → 𝑅 Er 𝑋 ) |
|
|
ertr3d.5 |
⊢ ( 𝜑 → 𝐵 𝑅 𝐴 ) |
|
|
ertr3d.6 |
⊢ ( 𝜑 → 𝐵 𝑅 𝐶 ) |
|
Assertion |
ertr3d |
⊢ ( 𝜑 → 𝐴 𝑅 𝐶 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
ersymb.1 |
⊢ ( 𝜑 → 𝑅 Er 𝑋 ) |
2 |
|
ertr3d.5 |
⊢ ( 𝜑 → 𝐵 𝑅 𝐴 ) |
3 |
|
ertr3d.6 |
⊢ ( 𝜑 → 𝐵 𝑅 𝐶 ) |
4 |
1 2
|
ersym |
⊢ ( 𝜑 → 𝐴 𝑅 𝐵 ) |
5 |
1 4 3
|
ertrd |
⊢ ( 𝜑 → 𝐴 𝑅 𝐶 ) |