Mathbox for Andrew Salmon < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm14.122c Structured version   Visualization version   GIF version

Theorem pm14.122c 37647
 Description: Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
Assertion
Ref Expression
pm14.122c (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem pm14.122c
StepHypRef Expression
1 pm14.122a 37645 . 2 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑)))
2 pm14.122b 37646 . 2 (𝐴𝑉 → ((∀𝑥(𝜑𝑥 = 𝐴) ∧ [𝐴 / 𝑥]𝜑) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
31, 2bitrd 267 1 (𝐴𝑉 → (∀𝑥(𝜑𝑥 = 𝐴) ↔ (∀𝑥(𝜑𝑥 = 𝐴) ∧ ∃𝑥𝜑)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator