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Theorem sbiota1 44453
Description: Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
sbiota1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Proof of Theorem sbiota1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eu6 2574 . . . 4 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
21biimpi 216 . . 3 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iota4 6542 . . 3 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
4 iotaval 6532 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
54eqcomd 2743 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6 spsbim 2072 . . . . . . . 8 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7 sbsbc 3792 . . . . . . . 8 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
8 sbsbc 3792 . . . . . . . 8 ([𝑦 / 𝑥]𝜓[𝑦 / 𝑥]𝜓)
96, 7, 83imtr3g 295 . . . . . . 7 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
10 dfsbcq 3790 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
11 dfsbcq 3790 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓[(℩𝑥𝜑) / 𝑥]𝜓))
1210, 11imbi12d 344 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
139, 12imbitrid 244 . . . . . 6 (𝑦 = (℩𝑥𝜑) → (∀𝑥(𝜑𝜓) → ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
1413com23 86 . . . . 5 (𝑦 = (℩𝑥𝜑) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
155, 14syl 17 . . . 4 (∀𝑥(𝜑𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
1615exlimiv 1930 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
172, 3, 16sylc 65 . 2 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓))
18 iotaexeu 44437 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
1910, 11anbi12d 632 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
2019imbi1d 341 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))))
21 sbcan 3838 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
22 spesbc 3882 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) → ∃𝑥(𝜑𝜓))
2321, 22sylbir 235 . . . . . . 7 (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))
2420, 23vtoclg 3554 . . . . . 6 ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)))
2524expd 415 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓))))
2618, 3, 25sylc 65 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓)))
2726anc2li 555 . . 3 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))))
28 eupicka 2634 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
2927, 28syl6 35 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑𝜓)))
3017, 29impbid 212 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wex 1779  [wsb 2064  wcel 2108  ∃!weu 2568  Vcvv 3480  [wsbc 3788  cio 6512
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 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-v 3482  df-sbc 3789  df-un 3956  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514
This theorem is referenced by:  sbaniota  44454
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