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Metamath Proof Explorer

Theorem sbaniota

Description: Theorem *14.26 in WhiteheadRussell p. 192. (Contributed by Andrew Salmon, 12-Jul-2011)

Ref Expression
Assertion sbaniota ⊢ ∃! x φ → ∃ x φ ∧ ψ ↔ [˙ ι x | φ / x]˙ ψ

Proof

Step Hyp Ref Expression
1 eupickbi ⊢ ∃! x φ → ∃ x φ ∧ ψ ↔ ∀ x φ → ψ
2 sbiota1 ⊢ ∃! x φ → ∀ x φ → ψ ↔ [˙ ι x | φ / x]˙ ψ
3 1 2 bitrd ⊢ ∃! x φ → ∃ x φ ∧ ψ ↔ [˙ ι x | φ / x]˙ ψ