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Theorem iotasbc 39557
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 3676 . 2 ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2 iotaexeu 39556 . . . . . . 7 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3 eueq 3588 . . . . . . 7 ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
42, 3sylib 210 . . . . . 6 (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5 eu6 2591 . . . . . . 7 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 iotaval 6110 . . . . . . . . . 10 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
76eqcomd 2783 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
87ancri 545 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
98eximi 1878 . . . . . . 7 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 9sylbi 209 . . . . . 6 (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
11 eupick 2662 . . . . . 6 ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
124, 10, 11syl2anc 579 . . . . 5 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
1312, 7impbid1 217 . . . 4 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1413anbi1d 623 . . 3 (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
1514exbidv 1964 . 2 (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
161, 15syl5bb 275 1 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  wal 1599   = wceq 1601  wex 1823  wcel 2106  ∃!weu 2585  Vcvv 3397  [wsbc 3651  cio 6097
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-rex 3095  df-v 3399  df-sbc 3652  df-un 3796  df-sn 4398  df-pr 4400  df-uni 4672  df-iota 6099
This theorem is referenced by:  iotasbc2  39558  iotavalb  39568  fvsb  39592
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