Theorem uniabio 4350

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		abbi 2464	. . . . 5 ⊢ (∀x(φ ↔ x = y) ↔ {x ∣ φ} = {x ∣ x = y})
2	1	biimpi 186	. . . 4 ⊢ (∀x(φ ↔ x = y) → {x ∣ φ} = {x ∣ x = y})
3		df-sn 3742	. . . 4 ⊢ {y} = {x ∣ x = y}
4	2, 3	syl6eqr 2403	. . 3 ⊢ (∀x(φ ↔ x = y) → {x ∣ φ} = {y})
5	4	unieqd 3903	. 2 ⊢ (∀x(φ ↔ x = y) → ∪{x ∣ φ} = ∪{y})
6		vex 2863	. . 3 ⊢ y ∈ V
7	6	unisn 3908	. 2 ⊢ ∪{y} = y
8	5, 7	syl6eq 2401	1 ⊢ (∀x(φ ↔ x = y) → ∪{x ∣ φ} = y)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642  {cab 2339  {csn 3738  ∪cuni 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893

This theorem is referenced by:  iotaval  4351  iotauni  4352