 Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege55lem2b Structured version   Visualization version   GIF version

Theorem frege55lem2b 37008
 Description: Lemma for frege55b 37009. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege55lem2b (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)

Proof of Theorem frege55lem2b
StepHypRef Expression
1 frege54cor1b 37006 . 2 [𝑥 / 𝑧]𝑧 = 𝑥
2 frege53b 37002 . 2 ([𝑥 / 𝑧]𝑧 = 𝑥 → (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥))
31, 2ax-mp 5 1 (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  [wsb 1865 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-12 2031  ax-13 2227  ax-ext 2584  ax-frege8 36921  ax-frege52c 37000 This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-sbc 3397 This theorem is referenced by:  frege55b  37009
 Copyright terms: Public domain W3C validator