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Metamath Proof Explorer

Theorem 19.41vvv

Description: Version of 19.41 with three quantifiers and a disjoint variable condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995)

Ref Expression
Assertion 19.41vvv ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ ∃ x ∃ y ∃ z φ ∧ ψ

Proof

Step Hyp Ref Expression
1 19.41vv ⊢ ∃ y ∃ z φ ∧ ψ ↔ ∃ y ∃ z φ ∧ ψ
2 1 exbii ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ ∃ x ∃ y ∃ z φ ∧ ψ
3 19.41v ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ ∃ x ∃ y ∃ z φ ∧ ψ
4 2 3 bitri ⊢ ∃ x ∃ y ∃ z φ ∧ ψ ↔ ∃ x ∃ y ∃ z φ ∧ ψ