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Metamath Proof Explorer

Theorem 19.42vvv

Description: Version of 19.42 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011) (Proof shortened by Wolf Lammen, 27-Aug-2023)

Ref Expression
Assertion 19.42vvv ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ φ ∧ ∃ x ∃ y ∃ z ψ

Proof

Step Hyp Ref Expression
1 exdistr2 ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ ∃ x φ ∧ ∃ y ∃ z ψ
2 19.42v ⊢ ∃ x φ ∧ ∃ y ∃ z ψ ↔ φ ∧ ∃ x ∃ y ∃ z ψ
3 1 2 bitri ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ φ ∧ ∃ x ∃ y ∃ z ψ