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Metamath Proof Explorer

Theorem mpteq2dfa

Description: Slightly more general equality inference for the maps-to notation. (Contributed by Glauco Siliprandi, 21-Dec-2024)

Ref Expression
Hypotheses mpteq2dfa.1 ⊢ Ⅎ 𝑥 𝜑
mpteq2dfa.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
Assertion mpteq2dfa ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 mpteq2dfa.1 ⊢ Ⅎ 𝑥 𝜑
2 mpteq2dfa.2 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 = 𝐶 )
3 1 2 mpteq2da ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) )