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∈ABφ↔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∈AB¬φ↔¬ψ∧¬χ
6 ralnex ⊢∀x∈AB¬φ↔¬∃x∈ABφ
7 pm4.56 ⊢¬ψ∧¬χ↔¬ψ∨χ
8 6 7 bibi12i ⊢∀x∈AB¬φ↔¬ψ∧¬χ↔¬∃x∈ABφ↔¬ψ∨χ
9 notbi ⊢∃x∈ABφ↔ψ∨χ↔¬∃x∈ABφ↔¬ψ∨χ
10 8 9 sylbb2 ⊢∀x∈AB¬φ↔¬ψ∧¬χ→∃x∈ABφ↔ψ∨χ
11 5 10 syl ⊢A∈V∧B∈W→∃x∈ABφ↔ψ∨χ