Theorem uniabio 6391

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.)

Assertion

Ref	Expression
uniabio	⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem uniabio

Step	Hyp	Ref	Expression
1		abbi1 2807	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
2		df-sn 4559	. . . 4 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	1, 2	eqtr4di 2797	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → {𝑥 ∣ 𝜑} = {𝑦})
4	3	unieqd 4850	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = ∪ {𝑦})
5		vex 3426	. . 3 ⊢ 𝑦 ∈ V
6	5	unisn 4858	. 2 ⊢ ∪ {𝑦} = 𝑦
7	4, 6	eqtrdi 2795	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑥 ∣ 𝜑} = 𝑦)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  {cab 2715  {csn 4558  ∪ cuni 4836

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-in 3890  df-ss 3900  df-sn 4559  df-pr 4561  df-uni 4837

This theorem is referenced by:  iotaval  6392  iotauni  6393  sn-iotaval  40119