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Theorem pm14.18 41052
 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 41042 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2 spsbc 3771 . 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 1536   ∈ wcel 2115  ∃!weu 2654  Vcvv 3480  [wsbc 3758  ℩cio 6300 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759  df-un 3924  df-in 3926  df-ss 3936  df-sn 4551  df-pr 4553  df-uni 4825  df-iota 6302 This theorem is referenced by: (None)
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