Theorem eceq1

Description: Equality theorem for equivalence class. (Contributed by NM, 23-Jul-1995)

	Ref	Expression
Assertion	eceq1	⊢ ( 𝐴 = 𝐵 → [ 𝐴 ] 𝐶 = [ 𝐵 ] 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sneq	⊢ ( 𝐴 = 𝐵 → { 𝐴 } = { 𝐵 } )
2	1	imaeq2d	⊢ ( 𝐴 = 𝐵 → ( 𝐶 “ { 𝐴 } ) = ( 𝐶 “ { 𝐵 } ) )
3		df-ec	⊢ [ 𝐴 ] 𝐶 = ( 𝐶 “ { 𝐴 } )
4		df-ec	⊢ [ 𝐵 ] 𝐶 = ( 𝐶 “ { 𝐵 } )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → [ 𝐴 ] 𝐶 = [ 𝐵 ] 𝐶 )