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 )