Metamath Proof Explorer
Description: Equality theorem for the A -coset and B -coset of C ,
deduction version. (Contributed by Peter Mazsa, 23-Apr-2021)
|
|
Ref |
Expression |
|
Hypothesis |
eceq2d.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
|
Assertion |
eceq2d |
⊢ ( 𝜑 → [ 𝐶 ] 𝐴 = [ 𝐶 ] 𝐵 ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
eceq2d.1 |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
2 |
|
eceq2 |
⊢ ( 𝐴 = 𝐵 → [ 𝐶 ] 𝐴 = [ 𝐶 ] 𝐵 ) |
3 |
1 2
|
syl |
⊢ ( 𝜑 → [ 𝐶 ] 𝐴 = [ 𝐶 ] 𝐵 ) |