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Theorem pm14.18 38449
 Description: Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
pm14.18 (∃!𝑥𝜑 → (∀𝑥𝜓[(℩𝑥𝜑) / 𝑥]𝜓))

Proof of Theorem pm14.18
StepHypRef Expression
1 iotaexeu 38439 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2 spsbc 3442 . 2 ((℩𝑥𝜑) ∈ V → (∀𝑥𝜓[(℩𝑥𝜑) / 𝑥]𝜓))
31, 2syl 17 1 (∃!𝑥𝜑 → (∀𝑥𝜓[(℩𝑥𝜑) / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1479   ∈ wcel 1988  ∃!weu 2468  Vcvv 3195  [wsbc 3429  ℩cio 5837 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-rex 2915  df-v 3197  df-sbc 3430  df-un 3572  df-sn 4169  df-pr 4171  df-uni 4428  df-iota 5839 This theorem is referenced by: (None)
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