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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