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Theorem iota4an 6330
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 6329 . 2 (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓))
2 iotaex 6328 . . . 4 (℩𝑥(𝜑𝜓)) ∈ V
3 simpl 485 . . . . 5 ((𝜑𝜓) → 𝜑)
43sbcth 3785 . . . 4 ((℩𝑥(𝜑𝜓)) ∈ V → [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑))
52, 4ax-mp 5 . . 3 [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑)
6 sbcimg 3818 . . . 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 232 . 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 208  wa 398  wcel 2107  ∃!weu 2647  Vcvv 3493  [wsbc 3770  cio 6305
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-nul 5201
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-sn 4560  df-pr 4562  df-uni 4831  df-iota 6307
This theorem is referenced by: (None)
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