This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem feq1dd

Description: Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses feq1dd.eq ⊢ ( 𝜑 → 𝐹 = 𝐺 )
feq1dd.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
Assertion feq1dd ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 )

Proof

Step Hyp Ref Expression
1 feq1dd.eq ⊢ ( 𝜑 → 𝐹 = 𝐺 )
2 feq1dd.f ⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 )
3 1 feq1d ⊢ ( 𝜑 → ( 𝐹 : 𝐴 ⟶ 𝐵 ↔ 𝐺 : 𝐴 ⟶ 𝐵 ) )
4 2 3 mpbid ⊢ ( 𝜑 → 𝐺 : 𝐴 ⟶ 𝐵 )