Theorem oveq1

Description: Equality theorem for operation value. (Contributed by NM, 28-Feb-1995)

	Ref	Expression
Assertion	oveq1	⊢ ( 𝐴 = 𝐵 → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		opeq1	⊢ ( 𝐴 = 𝐵 → ⟨ 𝐴 , 𝐶 ⟩ = ⟨ 𝐵 , 𝐶 ⟩ )
2	1	fveq2d	⊢ ( 𝐴 = 𝐵 → ( 𝐹 ‘ ⟨ 𝐴 , 𝐶 ⟩ ) = ( 𝐹 ‘ ⟨ 𝐵 , 𝐶 ⟩ ) )
3		df-ov	⊢ ( 𝐴 𝐹 𝐶 ) = ( 𝐹 ‘ ⟨ 𝐴 , 𝐶 ⟩ )
4		df-ov	⊢ ( 𝐵 𝐹 𝐶 ) = ( 𝐹 ‘ ⟨ 𝐵 , 𝐶 ⟩ )
5	2 3 4	3eqtr4g	⊢ ( 𝐴 = 𝐵 → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐶 ) )