Theorem oveq12d

Description: Equality deduction for operation value. (Contributed by NM, 13-Mar-1995) (Proof shortened by Andrew Salmon, 22-Oct-2011)

	Ref	Expression
Hypotheses	oveq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	oveq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	oveq12d	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		oveq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		oveq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3		oveq12	⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐷 ) )
4	1 2 3	syl2anc	⊢ ( 𝜑 → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐷 ) )