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Theorem iota4an 5787
Description: Theorem *14.23 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
iota4an (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)

Proof of Theorem iota4an
StepHypRef Expression
1 iota4 5786 . 2 (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓))
2 iotaex 5785 . . . 4 (℩𝑥(𝜑𝜓)) ∈ V
3 simpl 472 . . . . 5 ((𝜑𝜓) → 𝜑)
43sbcth 3417 . . . 4 ((℩𝑥(𝜑𝜓)) ∈ V → [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑))
52, 4ax-mp 5 . . 3 [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑)
6 sbcimg 3444 . . . 4 ((℩𝑥(𝜑𝜓)) ∈ V → ([(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑) ↔ ([(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)))
72, 6ax-mp 5 . . 3 ([(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑) ↔ ([(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑))
85, 7mpbi 219 . 2 ([(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)
91, 8syl 17 1 (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 195  wa 383  wcel 1977  ∃!weu 2458  Vcvv 3173  [wsbc 3402  cio 5766
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-iota 5768
This theorem is referenced by: (None)
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