Theorem iotaval 4351

Description: Theorem 8.19 in [Quine] p. 57. This theorem is the fundamental property of iota. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotaval	⊢ (∀x(φ ↔ x = y) → (℩xφ) = y)

Distinct variable group:   x,y

Allowed substitution hints:   φ(x,y)

Proof of Theorem iotaval

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiota2 4341	. 2 ⊢ (℩xφ) = ∪{z ∣ ∀x(φ ↔ x = z)}
2		vex 2863	. . . . . . 7 ⊢ y ∈ V
3		sbeqalb 3099	. . . . . . . 8 ⊢ (y ∈ V → ((∀x(φ ↔ x = y) ∧ ∀x(φ ↔ x = z)) → y = z))
4		equcomi 1679	. . . . . . . 8 ⊢ (y = z → z = y)
5	3, 4	syl6 29	. . . . . . 7 ⊢ (y ∈ V → ((∀x(φ ↔ x = y) ∧ ∀x(φ ↔ x = z)) → z = y))
6	2, 5	ax-mp 5	. . . . . 6 ⊢ ((∀x(φ ↔ x = y) ∧ ∀x(φ ↔ x = z)) → z = y)
7	6	ex 423	. . . . 5 ⊢ (∀x(φ ↔ x = y) → (∀x(φ ↔ x = z) → z = y))
8		equequ2 1686	. . . . . . . . . 10 ⊢ (y = z → (x = y ↔ x = z))
9	8	eqcoms 2356	. . . . . . . . 9 ⊢ (z = y → (x = y ↔ x = z))
10	9	bibi2d 309	. . . . . . . 8 ⊢ (z = y → ((φ ↔ x = y) ↔ (φ ↔ x = z)))
11	10	biimpd 198	. . . . . . 7 ⊢ (z = y → ((φ ↔ x = y) → (φ ↔ x = z)))
12	11	alimdv 1621	. . . . . 6 ⊢ (z = y → (∀x(φ ↔ x = y) → ∀x(φ ↔ x = z)))
13	12	com12 27	. . . . 5 ⊢ (∀x(φ ↔ x = y) → (z = y → ∀x(φ ↔ x = z)))
14	7, 13	impbid 183	. . . 4 ⊢ (∀x(φ ↔ x = y) → (∀x(φ ↔ x = z) ↔ z = y))
15	14	alrimiv 1631	. . 3 ⊢ (∀x(φ ↔ x = y) → ∀z(∀x(φ ↔ x = z) ↔ z = y))
16		uniabio 4350	. . 3 ⊢ (∀z(∀x(φ ↔ x = z) ↔ z = y) → ∪{z ∣ ∀x(φ ↔ x = z)} = y)
17	15, 16	syl 15	. 2 ⊢ (∀x(φ ↔ x = y) → ∪{z ∣ ∀x(φ ↔ x = z)} = y)
18	1, 17	syl5eq 2397	1 ⊢ (∀x(φ ↔ x = y) → (℩xφ) = y)

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2860  ∪cuni 3892  ℩cio 4338

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-sbc 3048  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-uni 3893  df-iota 4340

This theorem is referenced by:  iotauni  4352  iota1  4354  iotaex  4357  iota4  4358  iota5  4360