This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )