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Metamath Proof Explorer

Theorem eupickbi

Description: Theorem *14.26 in WhiteheadRussell p. 192. (Contributed by Andrew Salmon, 11-Jul-2011) (Proof shortened by Wolf Lammen, 27-Dec-2018)

Ref Expression
Assertion eupickbi ⊢ ∃! x φ → ∃ x φ ∧ ψ ↔ ∀ x φ → ψ

Proof

Step Hyp Ref Expression
1 eupicka ⊢ ∃! x φ ∧ ∃ x φ ∧ ψ → ∀ x φ → ψ
2 1 ex ⊢ ∃! x φ → ∃ x φ ∧ ψ → ∀ x φ → ψ
3 euex ⊢ ∃! x φ → ∃ x φ
4 exintr ⊢ ∀ x φ → ψ → ∃ x φ → ∃ x φ ∧ ψ
5 3 4 syl5com ⊢ ∃! x φ → ∀ x φ → ψ → ∃ x φ ∧ ψ
6 2 5 impbid ⊢ ∃! x φ → ∃ x φ ∧ ψ ↔ ∀ x φ → ψ