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Theorem dfiota2 4340
 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 4339 . 2 (℩xφ) = {y {x φ} = {y}}
2 df-sn 3741 . . . . . 6 {y} = {x x = y}
32eqeq2i 2363 . . . . 5 ({x φ} = {y} ↔ {x φ} = {x x = y})
4 abbi 2463 . . . . 5 (x(φx = y) ↔ {x φ} = {x x = y})
53, 4bitr4i 243 . . . 4 ({x φ} = {y} ↔ x(φx = y))
65abbii 2465 . . 3 {y {x φ} = {y}} = {y x(φx = y)}
76unieqi 3901 . 2 {y {x φ} = {y}} = {y x(φx = y)}
81, 7eqtri 2373 1 (℩xφ) = {y x(φx = y)}
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642  {cab 2339  {csn 3737  ∪cuni 3891  ℩cio 4337 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-sn 3741  df-uni 3892  df-iota 4339 This theorem is referenced by:  nfiota1  4341  nfiotad  4342  cbviota  4344  sb8iota  4346  iotaval  4350  iotanul  4354  eqtfinrelk  4486  fv2  5324  tcfnex  6244
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