Theorem iotasbc 40758

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 3802	. 2 ⊢ ([(℩𝑥𝜑) / 𝑦]𝜓 ↔ ∃𝑦(𝑦 = (℩𝑥𝜑) ∧ 𝜓))
2		iotaexeu 40757	. . . . . . 7 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
3		eueq 3701	. . . . . . 7 ⊢ ((℩𝑥𝜑) ∈ V ↔ ∃!𝑦 𝑦 = (℩𝑥𝜑))
4	2, 3	sylib 220	. . . . . 6 ⊢ (∃!𝑥𝜑 → ∃!𝑦 𝑦 = (℩𝑥𝜑))
5		eu6 2659	. . . . . . 7 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
6		iotaval 6331	. . . . . . . . . 10 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
7	6	eqcomd 2829	. . . . . . . . 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 1835	. . . . . . 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 2718	. . . . . 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 1922	. 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 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653  Vcvv 3496  [wsbc 3774  ℩cio 6314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-v 3498  df-sbc 3775  df-un 3943  df-in 3945  df-ss 3954  df-sn 4570  df-pr 4572  df-uni 4841  df-iota 6316

This theorem is referenced by:  iotasbc2  40759  iotavalb  40769  fvsb  40791