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Metamath Proof Explorer

Theorem dfiota2

Description: Alternate definition for descriptions. Definition 8.18 in Quine p. 56. (Contributed by Andrew Salmon, 30-Jun-2011)

Ref Expression
Assertion dfiota2 ⊢ ι x | φ = ⋃ y | ∀ x φ ↔ x = y

Proof

Step Hyp Ref Expression
1 df-iota ⊢ ι x | φ = ⋃ y | x | φ = y
2 absn ⊢ x | φ = y ↔ ∀ x φ ↔ x = y
3 2 abbii ⊢ y | x | φ = y = y | ∀ x φ ↔ x = y
4 3 unieqi ⊢ ⋃ y | x | φ = y = ⋃ y | ∀ x φ ↔ x = y
5 1 4 eqtri ⊢ ι x | φ = ⋃ y | ∀ x φ ↔ x = y