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Theorem iotaex 4356
 Description: Theorem 8.23 in [Quine] p. 58. This theorem proves the existence of the ℩ class under our definition. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotaex (℩xφ) V

Proof of Theorem iotaex
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 iotaval 4350 . . . . 5 (x(φx = z) → (℩xφ) = z)
21eqcomd 2358 . . . 4 (x(φx = z) → z = (℩xφ))
32eximi 1576 . . 3 (zx(φx = z) → z z = (℩xφ))
4 df-eu 2208 . . 3 (∃!xφzx(φx = z))
5 isset 2863 . . 3 ((℩xφ) V ↔ z z = (℩xφ))
63, 4, 53imtr4i 257 . 2 (∃!xφ → (℩xφ) V)
7 iotanul 4354 . . 3 ∃!xφ → (℩xφ) = )
8 0ex 4110 . . 3 V
97, 8syl6eqel 2441 . 2 ∃!xφ → (℩xφ) V)
106, 9pm2.61i 156 1 (℩xφ) V
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃!weu 2204  Vcvv 2859  ∅c0 3550  ℩cio 4337 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  ax-nin 4078 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-uni 3892  df-iota 4339 This theorem is referenced by:  iota4an  4358  ncfinex  4472  tfinex  4485  fvex  5339  tcex  6157
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