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Theorem iotasbc 42706
Description: Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define in terms of a function of (℩𝑥𝜑). Their definition differs in that a function of (℩𝑥𝜑) evaluates to "false" when there isn't a single 𝑥 that satisfies 𝜑. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
iotasbc (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
Distinct variable groups:   𝑥,𝑦   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem iotasbc
StepHypRef Expression
1 sbc5 3768 . 2 ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2 iotaexeu 42705 . . . . . . 7 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3 eueq 3667 . . . . . . 7 ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
42, 3sylib 217 . . . . . 6 (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5 eu6 2573 . . . . . . 7 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 iotaval 6468 . . . . . . . . . 10 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
76eqcomd 2743 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
87ancri 551 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
98eximi 1838 . . . . . . 7 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 9sylbi 216 . . . . . 6 (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
11 eupick 2634 . . . . . 6 ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
124, 10, 11syl2anc 585 . . . . 5 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
1312, 7impbid1 224 . . . 4 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1413anbi1d 631 . . 3 (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
1514exbidv 1925 . 2 (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
161, 15bitrid 283 1 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540   = wceq 1542  wex 1782  wcel 2107  ∃!weu 2567  Vcvv 3446  [wsbc 3740  cio 6447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3448  df-sbc 3741  df-un 3916  df-in 3918  df-ss 3928  df-sn 4588  df-pr 4590  df-uni 4867  df-iota 6449
This theorem is referenced by:  iotasbc2  42707  iotavalb  42717  fvsb  42739
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