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Metamath Proof Explorer

Theorem 19.42vv

Description: Version of 19.42 with two quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995)

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

Proof

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