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Theorem iota4an 6031
 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 6030 . 2 (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓))
2 iotaex 6029 . . . 4 (℩𝑥(𝜑𝜓)) ∈ V
3 simpl 474 . . . . 5 ((𝜑𝜓) → 𝜑)
43sbcth 3591 . . . 4 ((℩𝑥(𝜑𝜓)) ∈ V → [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑))
52, 4ax-mp 5 . . 3 [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑)
6 sbcimg 3618 . . . 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 220 . 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 196   ∧ wa 383   ∈ wcel 2139  ∃!weu 2607  Vcvv 3340  [wsbc 3576  ℩cio 6010 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-nul 4941 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-sn 4322  df-pr 4324  df-uni 4589  df-iota 6012 This theorem is referenced by: (None)
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