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Theorem iotasbc 41057
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 3775 . 2 ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2 iotaexeu 41056 . . . . . . 7 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3 eueq 3674 . . . . . . 7 ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
42, 3sylib 221 . . . . . 6 (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5 eu6 2658 . . . . . . 7 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 iotaval 6308 . . . . . . . . . 10 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
76eqcomd 2828 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
87ancri 553 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
98eximi 1836 . . . . . . 7 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 9sylbi 220 . . . . . 6 (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
11 eupick 2719 . . . . . 6 ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
124, 10, 11syl2anc 587 . . . . 5 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
1312, 7impbid1 228 . . . 4 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1413anbi1d 632 . . 3 (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
1514exbidv 1922 . 2 (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
161, 15syl5bb 286 1 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2114  ∃!weu 2652  Vcvv 3469  [wsbc 3747  cio 6291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-v 3471  df-sbc 3748  df-un 3913  df-in 3915  df-ss 3925  df-sn 4540  df-pr 4542  df-uni 4814  df-iota 6293
This theorem is referenced by:  iotasbc2  41058  iotavalb  41068  fvsb  41090
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