Theorem sbiota1 45172

Description: Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)

Assertion

Ref	Expression
sbiota1	⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Proof of Theorem sbiota1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eu6 2602	. . . 4 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2	1	biimpi 219	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3		iota4 6517	. . 3 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
4		iotaval 6510	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
5	4	eqcomd 2769	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6		spsbim 2106	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7		sbsbc 3748	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8		sbsbc 3748	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)
9	6, 7, 8	3imtr3g 298	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
10		dfsbcq 3746	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
11		dfsbcq 3746	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓 ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
12	10, 11	imbi12d 347	. . . . . . 7 ⊢ (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑 → [(℩𝑥𝜑) / 𝑥]𝜓)))
13	9, 12	imbitrid 247	. . . . . 6 ⊢ (𝑦 = (℩𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → ([(℩𝑥𝜑) / 𝑥]𝜑 → [(℩𝑥𝜑) / 𝑥]𝜓)))
14	13	com23 87	. . . . 5 ⊢ (𝑦 = (℩𝑥𝜑) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
15	5, 14	syl 18	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
16	15	exlimiv 1960	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
17	2, 3, 16	sylc 66	. 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓))
18		iotaexeu 45156	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
19	10, 11	anbi12d 643	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓)))
20	19	imbi1d 344	. . . . . . 7 ⊢ (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓))))
21		sbcan 3793	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
22		spesbc 3835	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
23	21, 22	sylbir 238	. . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
24	20, 23	vtoclg 3522	. . . . . 6 ⊢ ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
25	24	expd 420	. . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑 ∧ 𝜓))))
26	18, 3, 25	sylc 66	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
27	26	anc2li 564	. . 3 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))))
28		eupicka 2662	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
29	27, 28	syl6 36	. 2 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑 → 𝜓)))
30	17, 29	impbid 215	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   = wceq 1570  ∃wex 1809  [wsb 2096   ∈ wcel 2143  ∃!weu 2596  Vcvv 3455  [wsbc 3744  ℩cio 6490

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-v 3457  df-sbc 3745  df-un 3910  df-ss 3922  df-sn 4590  df-pr 4592  df-uni 4873  df-iota 6492

This theorem is used by:  sbaniota  45173