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Theorem sbiota1 37456
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 2457 . . . 4 (∃!𝑥𝜑 ↔ ∃𝑦𝑥(𝜑𝑥 = 𝑦))
21biimpi 204 . . 3 (∃!𝑥𝜑 → ∃𝑦𝑥(𝜑𝑥 = 𝑦))
3 iota4 5768 . . 3 (∃!𝑥𝜑[(℩𝑥𝜑) / 𝑥]𝜑)
4 iotaval 5761 . . . . . 6 (∀𝑥(𝜑𝑥 = 𝑦) → (℩𝑥𝜑) = 𝑦)
54eqcomd 2611 . . . . 5 (∀𝑥(𝜑𝑥 = 𝑦) → 𝑦 = (℩𝑥𝜑))
6 spsbim 2377 . . . . . . . 8 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
7 sbsbc 3401 . . . . . . . 8 ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜑)
8 sbsbc 3401 . . . . . . . 8 ([𝑦 / 𝑥]𝜓[𝑦 / 𝑥]𝜓)
96, 7, 83imtr3g 282 . . . . . . 7 (∀𝑥(𝜑𝜓) → ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
10 dfsbcq 3399 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜑))
11 dfsbcq 3399 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → ([𝑦 / 𝑥]𝜓[(℩𝑥𝜑) / 𝑥]𝜓))
1210, 11imbi12d 332 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
139, 12syl5ib 232 . . . . . 6 (𝑦 = (℩𝑥𝜑) → (∀𝑥(𝜑𝜓) → ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
1413com23 83 . . . . 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 1843 . . 3 (∃𝑦𝑥(𝜑𝑥 = 𝑦) → ([(℩𝑥𝜑) / 𝑥]𝜑 → (∀𝑥(𝜑𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓)))
172, 3, 16sylc 62 . 2 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) → [(℩𝑥𝜑) / 𝑥]𝜓))
18 iotaexeu 37440 . . . . 5 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
1910, 11anbi12d 742 . . . . . . . 8 (𝑦 = (℩𝑥𝜑) → (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) ↔ ([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓)))
2019imbi1d 329 . . . . . . 7 (𝑦 = (℩𝑥𝜑) → ((([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)) ↔ (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))))
21 sbcan 3440 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) ↔ ([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓))
22 spesbc 3482 . . . . . . . 8 ([𝑦 / 𝑥](𝜑𝜓) → ∃𝑥(𝜑𝜓))
2321, 22sylbir 223 . . . . . . 7 (([𝑦 / 𝑥]𝜑[𝑦 / 𝑥]𝜓) → ∃𝑥(𝜑𝜓))
2420, 23vtoclg 3234 . . . . . 6 ((℩𝑥𝜑) ∈ V → (([(℩𝑥𝜑) / 𝑥]𝜑[(℩𝑥𝜑) / 𝑥]𝜓) → ∃𝑥(𝜑𝜓)))
2524expd 450 . . . . 5 ((℩𝑥𝜑) ∈ V → ([(℩𝑥𝜑) / 𝑥]𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓))))
2618, 3, 25sylc 62 . . . 4 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∃𝑥(𝜑𝜓)))
2726anc2li 577 . . 3 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → (∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓))))
28 eupicka 2520 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → ∀𝑥(𝜑𝜓))
2927, 28syl6 34 . 2 (∃!𝑥𝜑 → ([(℩𝑥𝜑) / 𝑥]𝜓 → ∀𝑥(𝜑𝜓)))
3017, 29impbid 200 1 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  [wsb 1865  wcel 1975  ∃!weu 2453  Vcvv 3168  [wsbc 3397  cio 5748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ral 2896  df-rex 2897  df-v 3170  df-sbc 3398  df-un 3540  df-sn 4121  df-pr 4123  df-uni 4363  df-iota 5750
This theorem is referenced by:  sbaniota  37457
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