Theorem feq23

Description: Equality theorem for functions. (Contributed by FL, 14-Jul-2007) (Proof shortened by Andrew Salmon, 17-Sep-2011)

	Ref	Expression
Assertion	feq23	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐹 : 𝐶 ⟶ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		feq2	⊢ ( 𝐴 = 𝐶 → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐹 : 𝐶 ⟶ 𝐵 ) )
2		feq3	⊢ ( 𝐵 = 𝐷 → ( 𝐹 : 𝐶 ⟶ 𝐵 ↔ 𝐹 : 𝐶 ⟶ 𝐷 ) )
3	1 2	sylan9bb	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ) → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐹 : 𝐶 ⟶ 𝐷 ) )