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Theorem sbiota1 38952
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 df-eu 2502 . . . 4 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
21biimpi 206 . . 3 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iota4 5907 . . 3 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
4 iotaval 5900 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
54eqcomd 2657 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6 spsbim 2422 . . . . . . . 8 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7 sbsbc 3472 . . . . . . . 8 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
8 sbsbc 3472 . . . . . . . 8 ([𝑦 / 𝑥]𝜓[𝑦 / 𝑥]𝜓)
96, 7, 83imtr3g 284 . . . . . . 7 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
10 dfsbcq 3470 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
11 dfsbcq 3470 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓[(℩𝑥𝜑) / 𝑥]𝜓))
1210, 11imbi12d 333 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
139, 12syl5ib 234 . . . . . 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 1898 . . 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 38936 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
1910, 11anbi12d 747 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
2019imbi1d 330 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))))
21 sbcan 3511 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
22 spesbc 3554 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) → ∃𝑥(𝜑𝜓))
2321, 22sylbir 225 . . . . . . 7 (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))
2420, 23vtoclg 3297 . . . . . 6 ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)))
2524expd 451 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓))))
2618, 3, 25sylc 65 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓)))
2726anc2li 579 . . 3 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))))
28 eupicka 2566 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
2927, 28syl6 35 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑𝜓)))
3017, 29impbid 202 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  [wsb 1937  wcel 2030  ∃!weu 2498  Vcvv 3231  [wsbc 3468  cio 5887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-sbc 3469  df-un 3612  df-sn 4211  df-pr 4213  df-uni 4469  df-iota 5889
This theorem is referenced by:  sbaniota  38953
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