This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem uniabio

Description: Part of Theorem 8.17 in Quine p. 56. This theorem serves as a lemma for the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011)

Ref Expression
Assertion uniabio ⊢ ∀ x φ ↔ x = y → ⋃ x | φ = y

Proof

Step Hyp Ref Expression
1 abbi ⊢ ∀ x φ ↔ x = y → x | φ = x | x = y
2 df-sn ⊢ y = x | x = y
3 1 2 eqtr4di ⊢ ∀ x φ ↔ x = y → x | φ = y
4 3 unieqd ⊢ ∀ x φ ↔ x = y → ⋃ x | φ = ⋃ y
5 unisnv ⊢ ⋃ y = y
6 4 5 eqtrdi ⊢ ∀ x φ ↔ x = y → ⋃ x | φ = y