Theorem iotaval 5347

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 5336	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑧 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)}
2		vex 2824	. . . . . . 7 ⊢ 𝑦 ∈ V
3		sbeqalb 3108	. . . . . . . 8 ⊢ (𝑦 ∈ V → ((∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑧)) → 𝑦 = 𝑧))
4		equcomi 1756	. . . . . . . 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 115	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) → 𝑧 = 𝑦))
8		equequ2 1765	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
9	8	equcoms 1760	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
10	9	bibi2d 232	. . . . . . . 8 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑦) ↔ (𝜑 ↔ 𝑥 = 𝑧)))
11	10	biimpd 144	. . . . . . 7 ⊢ (𝑧 = 𝑦 → ((𝜑 ↔ 𝑥 = 𝑦) → (𝜑 ↔ 𝑥 = 𝑧)))
12	11	alimdv 1932	. . . . . 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 129	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
15	14	alrimiv 1927	. . 3 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∀𝑧(∀𝑥(𝜑 ↔ 𝑥 = 𝑧) ↔ 𝑧 = 𝑦))
16		uniabio 5346	. . 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	eqtrid 2283	1 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400   = wceq 1402   ∈ wcel 2209  {cab 2224  Vcvv 2821  ∪ cuni 3933  ℩cio 5333

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3714  df-pr 3715  df-uni 3934  df-iota 5335

This theorem is referenced by:  iotauni  5348  iota1  5350  euiotaex  5352  iota4  5355  iota5  5357