Theorem dfiota2 5287

Description: Alternate definition for descriptions. Definition 8.18 in [Quine] p. 56. (Contributed by Andrew Salmon, 30-Jun-2011.)

Assertion

Ref	Expression
dfiota2	⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfiota2

Step	Hyp	Ref	Expression
1		df-iota 5286	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		df-sn 3675	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	2	eqeq2i 2242	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
4		abbi 2345	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
5	3, 4	bitr4i 187	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6	5	abbii 2347	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
7	6	unieqi 3903	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
8	1, 7	eqtri 2252	1 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}

Syntax hints:   ↔ wb 105  ∀wal 1395   = wceq 1397  {cab 2217  {csn 3669  ∪ cuni 3893  ℩cio 5284

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rex 2516  df-sn 3675  df-uni 3894  df-iota 5286

This theorem is referenced by:  nfiota1  5288  nfiotadw  5289  cbviota  5291  sb8iota  5294  iotaval  5298  iotanul  5302  eliota  5314  fv2  5634