Metamath Proof Explorer

Theorem rexprg

Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011) (Revised by Mario Carneiro, 23-Apr-2015) Avoid ax-10 , ax-12 . (Revised by Gino Giotto, 30-Sep-2024)

Ref Expression
Hypotheses ralprg.1 ⊢ x = A → φ ↔ ψ
ralprg.2 ⊢ x = B → φ ↔ χ
Assertion rexprg ⊢ A ∈ V ∧ B ∈ W → ∃ x ∈ A B φ ↔ ψ ∨ χ

Proof

Step Hyp Ref Expression
1 ralprg.1 ⊢ x = A → φ ↔ ψ
2 ralprg.2 ⊢ x = B → φ ↔ χ
3 1 notbid ⊢ x = A → ¬ φ ↔ ¬ ψ
4 2 notbid ⊢ x = B → ¬ φ ↔ ¬ χ
5 3 4 ralprg ⊢ A ∈ V ∧ B ∈ W → ∀ x ∈ A B ¬ φ ↔ ¬ ψ ∧ ¬ χ
6 ralnex ⊢ ∀ x ∈ A B ¬ φ ↔ ¬ ∃ x ∈ A B φ
7 pm4.56 ⊢ ¬ ψ ∧ ¬ χ ↔ ¬ ψ ∨ χ
8 6 7 bibi12i ⊢ ∀ x ∈ A B ¬ φ ↔ ¬ ψ ∧ ¬ χ ↔ ¬ ∃ x ∈ A B φ ↔ ¬ ψ ∨ χ
9 notbi ⊢ ∃ x ∈ A B φ ↔ ψ ∨ χ ↔ ¬ ∃ x ∈ A B φ ↔ ¬ ψ ∨ χ
10 8 9 sylbb2 ⊢ ∀ x ∈ A B ¬ φ ↔ ¬ ψ ∧ ¬ χ → ∃ x ∈ A B φ ↔ ψ ∨ χ
11 5 10 syl ⊢ A ∈ V ∧ B ∈ W → ∃ x ∈ A B φ ↔ ψ ∨ χ