Theorem sbiota1 37657

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		df-eu 2462	. . . 4 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2	1	biimpi 205	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3		iota4 5786	. . 3 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
4		iotaval 5779	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
5	4	eqcomd 2616	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6		spsbim 2382	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7		sbsbc 3406	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8		sbsbc 3406	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)
9	6, 7, 8	3imtr3g 283	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
10		dfsbcq 3404	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
11		dfsbcq 3404	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓 ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
12	10, 11	imbi12d 333	. . . . . . 7 ⊢ (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑 → [(℩𝑥𝜑) / 𝑥]𝜓)))
13	9, 12	syl5ib 233	. . . . . 6 ⊢ (𝑦 = (℩𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → ([(℩𝑥𝜑) / 𝑥]𝜑 → [(℩𝑥𝜑) / 𝑥]𝜓)))
14	13	com23 84	. . . . 5 ⊢ (𝑦 = (℩𝑥𝜑) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
15	5, 14	syl 17	. . . 4 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
16	15	exlimiv 1845	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
17	2, 3, 16	sylc 63	. 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓))
18		iotaexeu 37641	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
19	10, 11	anbi12d 743	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓)))
20	19	imbi1d 330	. . . . . . 7 ⊢ (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓))))
21		sbcan 3445	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
22		spesbc 3487	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
23	21, 22	sylbir 224	. . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
24	20, 23	vtoclg 3239	. . . . . 6 ⊢ ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
25	24	expd 451	. . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑 ∧ 𝜓))))
26	18, 3, 25	sylc 63	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
27	26	anc2li 578	. . 3 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))))
28		eupicka 2525	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
29	27, 28	syl6 34	. 2 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑 → 𝜓)))
30	17, 29	impbid 201	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695  [wsb 1867   ∈ wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402  ℩cio 5766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-un 3545  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768

This theorem is referenced by:  sbaniota  37658