MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfiota2 Structured version   Visualization version   GIF version

Theorem dfiota2 6517
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(𝜑𝑥 = 𝑦)}
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem dfiota2
StepHypRef Expression
1 df-iota 6516 . 2 (℩𝑥𝜑) = {𝑦 ∣ {𝑥𝜑} = {𝑦}}
2 absn 4650 . . . 4 ({𝑥𝜑} = {𝑦} ↔ ∀𝑥(𝜑𝑥 = 𝑦))
32abbii 2807 . . 3 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
43unieqi 4924 . 2 {𝑦 ∣ {𝑥𝜑} = {𝑦}} = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
51, 4eqtri 2763 1 (℩𝑥𝜑) = {𝑦 ∣ ∀𝑥(𝜑𝑥 = 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wb 206  wal 1535   = wceq 1537  {cab 2712  {csn 4631   cuni 4912  cio 6514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-sn 4632  df-uni 4913  df-iota 6516
This theorem is referenced by:  nfiota1  6518  nfiotadw  6519  nfiotad  6521  cbviotaw  6523  cbviota  6525  sb8iota  6527  iotavalOLD  6537  iotanul  6541  fv2  6902
  Copyright terms: Public domain W3C validator