Theorem sbiota1 44458

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 2573	. . . 4 ⊢ (∃!𝑥𝜑 ↔ ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
2	1	biimpi 216	. . 3 ⊢ (∃!𝑥𝜑 → ∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦))
3		iota4 6512	. . 3 ⊢ (∃!𝑥𝜑 → [(℩𝑥𝜑) / 𝑥]𝜑)
4		iotaval 6502	. . . . . 6 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
5	4	eqcomd 2741	. . . . 5 ⊢ (∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6		spsbim 2072	. . . . . . . 8 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7		sbsbc 3769	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
8		sbsbc 3769	. . . . . . . 8 ⊢ ([𝑦 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓)
9	6, 7, 8	3imtr3g 295	. . . . . . 7 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
10		dfsbcq 3767	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑 ↔ [(℩𝑥𝜑) / 𝑥]𝜑))
11		dfsbcq 3767	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓 ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
12	10, 11	imbi12d 344	. . . . . . 7 ⊢ (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑 → [(℩𝑥𝜑) / 𝑥]𝜓)))
13	9, 12	imbitrid 244	. . . . . 6 ⊢ (𝑦 = (℩𝑥𝜑) → (∀𝑥(𝜑 → 𝜓) → ([(℩𝑥𝜑) / 𝑥]𝜑 → [(℩𝑥𝜑) / 𝑥]𝜓)))
14	13	com23 86	. . . . 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 1930	. . 3 ⊢ (∃𝑦∀𝑥(𝜑 ↔ 𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
17	2, 3, 16	sylc 65	. 2 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓))
18		iotaexeu 44442	. . . . 5 ⊢ (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
19	10, 11	anbi12d 632	. . . . . . . 8 ⊢ (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓)))
20	19	imbi1d 341	. . . . . . 7 ⊢ (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓))))
21		sbcan 3815	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓))
22		spesbc 3857	. . . . . . . 8 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
23	21, 22	sylbir 235	. . . . . . 7 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
24	20, 23	vtoclg 3533	. . . . . 6 ⊢ ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑 ∧ [(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑 ∧ 𝜓)))
25	24	expd 415	. . . . 5 ⊢ ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑 ∧ 𝜓))))
26	18, 3, 25	sylc 65	. . . 4 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑 ∧ 𝜓)))
27	26	anc2li 555	. . 3 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓))))
28		eupicka 2633	. . 3 ⊢ ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑 ∧ 𝜓)) → ∀𝑥(𝜑 → 𝜓))
29	27, 28	syl6 35	. 2 ⊢ (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑 → 𝜓)))
30	17, 29	impbid 212	1 ⊢ (∃!𝑥𝜑 → (∀𝑥(𝜑 → 𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779  [wsb 2064   ∈ wcel 2108  ∃!weu 2567  Vcvv 3459  [wsbc 3765  ℩cio 6482

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-v 3461  df-sbc 3766  df-un 3931  df-ss 3943  df-sn 4602  df-pr 4604  df-uni 4884  df-iota 6484

This theorem is referenced by:  sbaniota  44459