Theorem iotasbc 45594

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 3770	. 2 ⊢ ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2		iotaexeu 45593	. . . . . . 7 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3		eueq 3669	. . . . . . 7 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
4	2, 3	sylib 221	. . . . . 6 ⊢ (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5		eu6 2602	. . . . . . 7 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6		iotaval 6530	. . . . . . . . . 10 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
7	6	eqcomd 2769	. . . . . . . . 9 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
8	7	ancri 559	. . . . . . . 8 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
9	8	eximi 1868	. . . . . . 7 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
10	5, 9	sylbi 220	. . . . . 6 ⊢ (∃!𝑥𝜑 → ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
11		eupick 2661	. . . . . 6 ⊢ ((∃!𝑦 𝑦 = (℩𝑥𝜑) ∧ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦))) → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
12	4, 10, 11	syl2anc 596	. . . . 5 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) → ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
13	12, 7	impbid1 228	. . . 4 ⊢ (∃!𝑥𝜑 → (𝑦 = (℩𝑥𝜑) ↔ ∀𝑥(𝜑 ↔ 𝑥 = 𝑦)))
14	13	anbi1d 643	. . 3 ⊢ (∃!𝑥𝜑 → ((𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))
15	14	exbidv 1956	. 2 ⊢ (∃!𝑥𝜑 → (∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓) ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))
16	1, 15	bitrid 286	1 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(∀𝑥(𝜑 ↔ 𝑥 = 𝑦) ∧ 𝜓)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147  ∃!weu 2596  Vcvv 3454  [wsbc 3742  ℩cio 6510

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743  df-ss 3901  df-un 4154  df-sn 4589  df-pr 4591  df-uni 4871  df-iota 6512

This theorem is used by:  iotasbc2  45595  iotavalb  45605  fvsb  45625