Theorem iotaval 5164

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 5154	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
2		vex 2729	. . . . . . 7 ⊢ 𝑦 ∈ V
3		sbeqalb 3007	. . . . . . . 8 ⊢ (𝑦 ∈ V → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧))
4		equcomi 1692	. . . . . . . 8 ⊢ (𝑦 = 𝑧 → 𝑧 = 𝑦)
5	3, 4	syl6 33	. . . . . . 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 114	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = 𝑦))
8		equequ2 1701	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
9	8	equcoms 1696	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
10	9	bibi2d 231	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
11	10	biimpd 143	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 ↔ 𝑥 = 𝑧)))
12	11	alimdv 1867	. . . . . 6 ⊢ (𝑧 = 𝑦 → (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
13	12	com12 30	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑧 = 𝑦 → ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)))
14	7, 13	impbid 128	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
15	14	alrimiv 1862	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
16		uniabio 5163	. . 3 ⊢ (∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = 𝑦)
17	15, 16	syl 14	. 2 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)} = 𝑦)
18	1, 17	syl5eq 2211	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1341   = wceq 1343   ∈ wcel 2136  {cab 2151  Vcvv 2726  ∪ cuni 3789  ℩cio 5151

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153

This theorem is referenced by:  iotauni  5165  iota1  5167  euiotaex  5169  iota4  5171  iota5  5173