Theorem eceq1d

Description: Equality theorem for equivalence class (deduction form). (Contributed by Jim Kingdon, 31-Dec-2019)

	Ref	Expression
Hypothesis	eceq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	eceq1d	⊢ ( 𝜑 → [ 𝐴 ] 𝐶 = [ 𝐵 ] 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		eceq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		eceq1	⊢ ( 𝐴 = 𝐵 → [ 𝐴 ] 𝐶 = [ 𝐵 ] 𝐶 )
3	1 2	syl	⊢ ( 𝜑 → [ 𝐴 ] 𝐶 = [ 𝐵 ] 𝐶 )