 Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege55lem2c Structured version   Visualization version   GIF version

Theorem frege55lem2c 38713
 Description: Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55lem2c (𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥)
Distinct variable group:   𝑥,𝑧
Allowed substitution hints:   𝐴(𝑥,𝑧)

Proof of Theorem frege55lem2c
StepHypRef Expression
1 vex 3343 . . 3 𝑥 ∈ V
21frege54cor1c 38711 . 2 [𝑥 / 𝑧]𝑧 = 𝑥
3 frege53c 38710 . 2 ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥))
42, 3ax-mp 5 1 (𝑥 = 𝐴[𝐴 / 𝑧]𝑧 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632  Vcvv 3340  [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-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-frege8 38605  ax-frege52c 38684 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-sbc 3577  df-sn 4322 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator