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Theorem iota4an 5773
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 5772 . 2 (∃!𝑥(𝜑𝜓) → [(℩𝑥(𝜑𝜓)) / 𝑥](𝜑𝜓))
2 iotaex 5771 . . . 4 (℩𝑥(𝜑𝜓)) ∈ V
3 simpl 471 . . . . 5 ((𝜑𝜓) → 𝜑)
43sbcth 3416 . . . 4 ((℩𝑥(𝜑𝜓)) ∈ V → [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑))
52, 4ax-mp 5 . . 3 [(℩𝑥(𝜑𝜓)) / 𝑥]((𝜑𝜓) → 𝜑)
6 sbcimg 3443 . . . 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 218 . 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 194  wa 382  wcel 1976  ∃!weu 2457  Vcvv 3172  [wsbc 3401  cio 5752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-nul 4712
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 1867  df-eu 2461  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-sn 4125  df-pr 4127  df-uni 4367  df-iota 5754
This theorem is referenced by: (None)
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