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Metamath Proof Explorer

Theorem mpteq2da

Description: Slightly more general equality inference for the maps-to notation. (Contributed by FL, 14-Sep-2013) (Revised by Mario Carneiro, 16-Dec-2013) (Proof shortened by SN, 11-Nov-2024)

Ref Expression
Hypotheses mpteq2da.1 ⊢ Ⅎ x φ
mpteq2da.2 ⊢ φ ∧ x ∈ A → B = C
Assertion mpteq2da ⊢ φ → x ∈ A ⟼ B = x ∈ A ⟼ C

Proof

Step Hyp Ref Expression
1 mpteq2da.1 ⊢ Ⅎ x φ
2 mpteq2da.2 ⊢ φ ∧ x ∈ A → B = C
3 eqidd ⊢ φ → A = A
4 1 3 2 mpteq12da ⊢ φ → x ∈ A ⟼ B = x ∈ A ⟼ C