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Theorem pm14.18 37449
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 37439 . 2 (∃!𝑥𝜑 → (℩𝑥𝜑) ∈ V)
2 spsbc 3409 . 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 1472  wcel 1975  ∃!weu 2452  Vcvv 3167  [wsbc 3396  cio 5747
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584
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 1866  df-eu 2456  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-rex 2896  df-v 3169  df-sbc 3397  df-un 3539  df-sn 4120  df-pr 4122  df-uni 4362  df-iota 5749
This theorem is referenced by: (None)
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