Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  pm13.14 Structured version   Visualization version   GIF version

Theorem pm13.14 39130
Description: Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
Assertion
Ref Expression
pm13.14 (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥𝐴)

Proof of Theorem pm13.14
StepHypRef Expression
1 sbceq1a 3587 . . . 4 (𝑥 = 𝐴 → (𝜑[𝐴 / 𝑥]𝜑))
21biimprcd 240 . . 3 ([𝐴 / 𝑥]𝜑 → (𝑥 = 𝐴𝜑))
32necon3bd 2946 . 2 ([𝐴 / 𝑥]𝜑 → (¬ 𝜑𝑥𝐴))
43imp 444 1 (([𝐴 / 𝑥]𝜑 ∧ ¬ 𝜑) → 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wne 2932  [wsbc 3576
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-12 2196  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ne 2933  df-sbc 3577
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator