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 ( 𝜑𝑥 = 𝑦 ) → { 𝑥𝜑 } = 𝑦 )

Proof

Step Hyp Ref Expression
1 abbi1 ( ∀ 𝑥 ( 𝜑𝑥 = 𝑦 ) → { 𝑥𝜑 } = { 𝑥𝑥 = 𝑦 } )
2 df-sn { 𝑦 } = { 𝑥𝑥 = 𝑦 }
3 1 2 syl6eqr ( ∀ 𝑥 ( 𝜑𝑥 = 𝑦 ) → { 𝑥𝜑 } = { 𝑦 } )
4 3 unieqd ( ∀ 𝑥 ( 𝜑𝑥 = 𝑦 ) → { 𝑥𝜑 } = { 𝑦 } )
5 vex 𝑦 ∈ V
6 5 unisn { 𝑦 } = 𝑦
7 4 6 syl6eq ( ∀ 𝑥 ( 𝜑𝑥 = 𝑦 ) → { 𝑥𝜑 } = 𝑦 )