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Metamath Proof Explorer

Theorem feq12i

Description: Equality inference for functions. (Contributed by AV, 7-Feb-2021)

Ref Expression
Hypotheses feq12i.1 ⊢ 𝐹 = 𝐺
feq12i.2 ⊢ 𝐴 = 𝐵
Assertion feq12i ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 )

Proof

Step Hyp Ref Expression
1 feq12i.1 ⊢ 𝐹 = 𝐺
2 feq12i.2 ⊢ 𝐴 = 𝐵
3 eqid ⊢ 𝐶 = 𝐶
4 feq123 ⊢ ( ( 𝐹 = 𝐺 ∧ 𝐴 = 𝐵 ∧ 𝐶 = 𝐶 ) → ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 ) )
5 1 2 3 4 mp3an ⊢ ( 𝐹 : 𝐴 ⟶ 𝐶 ↔ 𝐺 : 𝐵 ⟶ 𝐶 )