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Theorem iotasbc 39039
 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 3566 . 2 ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2 iotaexeu 39038 . . . . . . 7 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3 eueq 3484 . . . . . . 7 ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
42, 3sylib 208 . . . . . 6 (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5 df-eu 2575 . . . . . . 7 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 iotaval 5975 . . . . . . . . . 10 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
76eqcomd 2730 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
87ancri 576 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
98eximi 1875 . . . . . . 7 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 9sylbi 207 . . . . . 6 (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
11 eupick 2638 . . . . . 6 ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
124, 10, 11syl2anc 696 . . . . 5 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
1312, 7impbid1 215 . . . 4 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1413anbi1d 743 . . 3 (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
1514exbidv 1963 . 2 (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
161, 15syl5bb 272 1 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1594   = wceq 1596  ∃wex 1817   ∈ wcel 2103  ∃!weu 2571  Vcvv 3304  [wsbc 3541  ℩cio 5962 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-rex 3020  df-v 3306  df-sbc 3542  df-un 3685  df-sn 4286  df-pr 4288  df-uni 4545  df-iota 5964 This theorem is referenced by:  iotasbc2  39040  iotavalb  39050  fvsb  39075
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