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Theorem iotasbc 38620
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 3460 . 2 ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2 iotaexeu 38619 . . . . . . 7 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3 eueq 3378 . . . . . . 7 ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
42, 3sylib 208 . . . . . 6 (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5 df-eu 2474 . . . . . . 7 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
6 iotaval 5862 . . . . . . . . . 10 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
76eqcomd 2628 . . . . . . . . 9 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
87ancri 575 . . . . . . . 8 (∀𝑥(𝜑𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
98eximi 1762 . . . . . . 7 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
105, 9sylbi 207 . . . . . 6 (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦)))
11 eupick 2536 . . . . . 6 ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
124, 10, 11syl2anc 693 . . . . 5 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑𝑥 = 𝑦)))
1312, 7impbid1 215 . . . 4 (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑𝑥 = 𝑦)))
1413anbi1d 741 . . 3 (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑𝑥 = 𝑦) ∧ 𝜓)))
1514exbidv 1850 . 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 384  wal 1481   = wceq 1483  wex 1704  wcel 1990  ∃!weu 2470  Vcvv 3200  [wsbc 3435  cio 5849
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851
This theorem is referenced by:  iotasbc2  38621  iotavalb  38631  fvsb  38656
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