Theorem coeq12d

Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012)

	Ref	Expression
Hypotheses	coeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	coeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	coeq12d	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		coeq12d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		coeq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	coeq1d	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐶 ) )
4	2	coeq2d	⊢ ( 𝜑 → ( 𝐵 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐷 ) )
5	3 4	eqtrd	⊢ ( 𝜑 → ( 𝐴 ∘ 𝐶 ) = ( 𝐵 ∘ 𝐷 ) )