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Metamath Proof Explorer

Theorem preqr1g

Description: Reverse equality lemma for unordered pairs. If two unordered pairs have the same second element, the first elements are equal. Closed form of preqr1 . (Contributed by AV, 29-Jan-2021) (Revised by AV, 18-Sep-2021)

Ref Expression
Assertion preqr1g ⊢ A ∈ V ∧ B ∈ W → A C = B C → A = B

Proof

Step Hyp Ref Expression
1 simpl ⊢ A ∈ V ∧ B ∈ W → A ∈ V
2 simpr ⊢ A ∈ V ∧ B ∈ W → B ∈ W
3 1 2 preq1b ⊢ A ∈ V ∧ B ∈ W → A C = B C ↔ A = B
4 3 biimpd ⊢ A ∈ V ∧ B ∈ W → A C = B C → A = B