Theorem dfaiota2 41977

Description: Alternate definition of the alternate version of Russell's definition of a description binder. Definition 8.18 in [Quine] p. 56. (Contributed by AV, 24-Aug-2022.)

Assertion

Ref	Expression
dfaiota2	⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfaiota2

Step	Hyp	Ref	Expression
1		df-aiota 41976	. 2 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}}
2		absn 4417	. . . 4 ⊢ ({𝑥 ∣ 𝜑} = {𝑦} ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3	2	abbii 2944	. . 3 ⊢ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
4	3	inteqi 4703	. 2 ⊢ ∩ {𝑦 ∣ {𝑥 ∣ 𝜑} = {𝑦}} = ∩ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}
5	1, 4	eqtri 2849	1 ⊢ (℩'𝑥𝜑) = ∩ {𝑦 ∣ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)}

Syntax hints:   ↔ wb 198  ∀wal 1654   = wceq 1656  {cab 2811  {csn 4399  ∩ cint 4699  ℩'caiota 41974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-sn 4400  df-int 4700  df-aiota 41976

This theorem is referenced by: (None)