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Metamath Proof Explorer

Theorem feq3dd

Description: Equality deduction for functions. (Contributed by Thierry Arnoux, 27-May-2025)

Ref Expression
Hypotheses feq3dd.eq ⊢ ( 𝜑 → 𝐵 = 𝐶 )
feq3dd.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
Assertion feq3dd ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )

Proof

Step Hyp Ref Expression
1 feq3dd.eq ⊢ ( 𝜑 → 𝐵 = 𝐶 )
2 feq3dd.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
3 1 feq3d ⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐹 : 𝐴 ⟶ 𝐶 ) )
4 2 3 mpbid ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐶 )