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Theorem sbiota1 43902
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 2563 . . . 4 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
21biimpi 215 . . 3 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iota4 6534 . . 3 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
4 iotaval 6524 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
54eqcomd 2734 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6 spsbim 2067 . . . . . . . 8 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7 sbsbc 3782 . . . . . . . 8 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
8 sbsbc 3782 . . . . . . . 8 ([𝑦 / 𝑥]𝜓[𝑦 / 𝑥]𝜓)
96, 7, 83imtr3g 294 . . . . . . 7 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
10 dfsbcq 3780 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
11 dfsbcq 3780 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓[(℩𝑥𝜑) / 𝑥]𝜓))
1210, 11imbi12d 343 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
139, 12imbitrid 243 . . . . . 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 1925 . . 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 43886 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
1910, 11anbi12d 630 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
2019imbi1d 340 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))))
21 sbcan 3831 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
22 spesbc 3877 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) → ∃𝑥(𝜑𝜓))
2321, 22sylbir 234 . . . . . . 7 (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))
2420, 23vtoclg 3542 . . . . . 6 ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)))
2524expd 414 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓))))
2618, 3, 25sylc 65 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓)))
2726anc2li 554 . . 3 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))))
28 eupicka 2625 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
2927, 28syl6 35 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑𝜓)))
3017, 29impbid 211 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1531   = wceq 1533  wex 1773  [wsb 2059  wcel 2098  ∃!weu 2557  Vcvv 3473  [wsbc 3778  cio 6503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-v 3475  df-sbc 3779  df-un 3954  df-in 3956  df-ss 3966  df-sn 4633  df-pr 4635  df-uni 4913  df-iota 6505
This theorem is referenced by:  sbaniota  43903
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