Theorem oveq123d

Description: Equality deduction for operation value. (Contributed by FL, 22-Dec-2008)

	Ref	Expression
Hypotheses	oveq123d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
	oveq123d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	oveq123d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	oveq123d	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐺 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		oveq123d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
2		oveq123d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		oveq123d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
4	1	oveqd	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐴 𝐺 𝐶 ) )
5	2 3	oveq12d	⊢ ( 𝜑 → ( 𝐴 𝐺 𝐶 ) = ( 𝐵 𝐺 𝐷 ) )
6	4 5	eqtrd	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐺 𝐷 ) )