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Theorem iota4an 6543
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 6542 . 2 (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓))
2 iotaex 6534 . . . 4 (℩𝑥(𝜑𝜓)) ∈ V
3 simpl 482 . . . . 5 ((𝜑𝜓) → 𝜑)
43sbcth 3803 . . . 4 ((℩𝑥(𝜑𝜓)) ∈ V → [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑))
52, 4ax-mp 5 . . 3 [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑)
6 sbcimg 3837 . . . 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 230 . 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 206  wa 395  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-12 2177  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  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-ne 2941  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-iota 6514
This theorem is referenced by: (None)
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