This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem eceq2d

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 )