Theorem iotasbc 40744

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(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))

Distinct variable groups:   𝑥,𝑦   𝜑,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem iotasbc

Step	Hyp	Ref	Expression
1		sbc5 3800	. 2 ⊢ ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2		iotaexeu 40743	. . . . . . 7 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3		eueq 3699	. . . . . . 7 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
4	2, 3	sylib 220	. . . . . 6 ⊢ (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5		eu6 2655	. . . . . . 7 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6		iotaval 6324	. . . . . . . . . 10 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
7	6	eqcomd 2827	. . . . . . . . 9 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
8	7	ancri 552	. . . . . . . 8 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
9	8	eximi 1831	. . . . . . 7 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	5, 9	sylbi 219	. . . . . 6 ⊢ (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
11		eupick 2714	. . . . . 6 ⊢ ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
12	4, 10, 11	syl2anc 586	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	12, 7	impbid1 227	. . . 4 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
14	13	anbi1d 631	. . 3 ⊢ (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))
15	14	exbidv 1918	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))
16	1, 15	syl5bb 285	1 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  ∃!weu 2649  Vcvv 3495  [wsbc 3772  ℩cio 6307

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rex 3144  df-v 3497  df-sbc 3773  df-un 3941  df-sn 4562  df-pr 4564  df-uni 4833  df-iota 6309

This theorem is referenced by:  iotasbc2  40745  iotavalb  40755  fvsb  40777