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Theorem iotanul 4354
 Description: Theorem 8.22 in [Quine] p. 57. This theorem is the result if there isn't exactly one x that satisfies φ. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotanul ∃!xφ → (℩xφ) = )

Proof of Theorem iotanul
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-eu 2208 . 2 (∃!xφzx(φx = z))
2 dfiota2 4340 . . 3 (℩xφ) = {z x(φx = z)}
3 alnex 1543 . . . . . 6 (z ¬ x(φx = z) ↔ ¬ zx(φx = z))
4 ax-1 5 . . . . . . . . . . 11 x(φx = z) → (z = z → ¬ x(φx = z)))
5 eqidd 2354 . . . . . . . . . . 11 x(φx = z) → z = z)
64, 5impbid1 194 . . . . . . . . . 10 x(φx = z) → (z = z ↔ ¬ x(φx = z)))
76con2bid 319 . . . . . . . . 9 x(φx = z) → (x(φx = z) ↔ ¬ z = z))
87alimi 1559 . . . . . . . 8 (z ¬ x(φx = z) → z(x(φx = z) ↔ ¬ z = z))
9 abbi 2463 . . . . . . . 8 (z(x(φx = z) ↔ ¬ z = z) ↔ {z x(φx = z)} = {z ¬ z = z})
108, 9sylib 188 . . . . . . 7 (z ¬ x(φx = z) → {z x(φx = z)} = {z ¬ z = z})
11 dfnul2 3552 . . . . . . 7 = {z ¬ z = z}
1210, 11syl6eqr 2403 . . . . . 6 (z ¬ x(φx = z) → {z x(φx = z)} = )
133, 12sylbir 204 . . . . 5 zx(φx = z) → {z x(φx = z)} = )
1413unieqd 3902 . . . 4 zx(φx = z) → {z x(φx = z)} = )
15 uni0 3918 . . . 4 =
1614, 15syl6eq 2401 . . 3 zx(φx = z) → {z x(φx = z)} = )
172, 16syl5eq 2397 . 2 zx(φx = z) → (℩xφ) = )
181, 17sylnbi 297 1 ∃!xφ → (℩xφ) = )
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642  ∃!weu 2204  {cab 2339  ∅c0 3550  ∪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-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-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-uni 3892  df-iota 4339 This theorem is referenced by:  iotassuni  4355  iotaex  4356  dfiota3  4370  dfiota4  4372  fvprc  5325
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