Theorem feq123d

Description: Equality deduction for functions. (Contributed by Paul Chapman, 22-Jun-2011)

	Ref	Expression
Hypotheses	feq12d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
	feq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	feq123d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	feq123d	⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		feq12d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
2		feq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		feq123d.3	⊢ ( 𝜑 → 𝐶 = 𝐷 )
4	1 2	feq12d	⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 ) )
5	3	feq3d	⊢ ( 𝜑 → ( 𝐺 : 𝐵 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐷 ) )
6	4 5	bitrd	⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐷 ) )