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Metamath Proof Explorer

Theorem oveqan12d

Description: Equality deduction for operation value. (Contributed by NM, 10-Aug-1995)

Ref Expression
Hypotheses oveq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
opreqan12i.2 ⊢ ( 𝜓 → 𝐶 = 𝐷 )
Assertion oveqan12d ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐷 ) )

Proof

Step Hyp Ref Expression
1 oveq1d.1 ⊢ ( 𝜑 → 𝐴 = 𝐵 )
2 opreqan12i.2 ⊢ ( 𝜓 → 𝐶 = 𝐷 )
3 oveq12 ⊢ ( ( 𝐴 = 𝐵 ∧ 𝐶 = 𝐷 ) → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐷 ) )
4 1 2 3 syl2an ⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝐴 𝐹 𝐶 ) = ( 𝐵 𝐹 𝐷 ) )