Mathbox for Andrew Salmon < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbaniota Structured version   Visualization version   GIF version

Theorem sbaniota 40757
 Description: Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
Assertion
Ref Expression
sbaniota (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))

Proof of Theorem sbaniota
StepHypRef Expression
1 eupickbi 2715 . 2 (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ ∀𝑥(𝜑𝜓)))
2 sbiota1 40756 . 2 (∃!𝑥𝜑 → (∀𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
31, 2bitrd 281 1 (∃!𝑥𝜑 → (∃𝑥(𝜑𝜓) ↔ [(℩𝑥𝜑) / 𝑥]𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1529  ∃wex 1774  ∃!weu 2647  [wsbc 3770  ℩cio 6305 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-v 3495  df-sbc 3771  df-un 3939  df-sn 4560  df-pr 4562  df-uni 4831  df-iota 6307 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator