Theorem feq12d

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 )
Assertion	feq12d	⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		feq12d.1	⊢ ( 𝜑 → 𝐹 = 𝐺 )
2		feq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3	1	feq1d	⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐴 ⟶ 𝐶 ) )
4	2	feq2d	⊢ ( 𝜑 → ( 𝐺 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 ) )