Theorem dfiota2 5769

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 5768	. 2 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		df-sn 4126	. . . . . 6 ⊢ {𝑦} = {𝑥 ∣ 𝑥 = 𝑦}
3	2	eqeq2i 2622	. . . . 5 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
4		abbi 2724	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝑥 = 𝑦})
5	3, 4	bitr4i 266	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6	5	abbii 2726	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
7	6	unieqi 4381	. 2 ⊢ ∪ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
8	1, 7	eqtri 2632	1 ⊢ (℩𝑥𝜑) = ∪ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}

Syntax hints:   ↔ wb 195  ∀wal 1473   = wceq 1475  {cab 2596  {csn 4125  ∪ cuni 4372  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-sn 4126  df-uni 4373  df-iota 5768

This theorem is referenced by:  nfiota1  5770  nfiotad  5771  cbviota  5773  sb8iota  5775  iotaval  5779  iotanul  5783  fv2  6098