Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege122d Structured version   Visualization version   GIF version

Theorem frege122d 40448
 Description: If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 40673. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege122d.a (𝜑𝐴 = (𝐹𝑋))
frege122d.b (𝜑𝐵 = (𝐹𝑋))
Assertion
Ref Expression
frege122d (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Proof of Theorem frege122d
StepHypRef Expression
1 frege122d.a . . 3 (𝜑𝐴 = (𝐹𝑋))
2 frege122d.b . . 3 (𝜑𝐵 = (𝐹𝑋))
31, 2eqtr4d 2839 . 2 (𝜑𝐴 = 𝐵)
43olcd 871 1 (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 844   = wceq 1538   class class class wbr 5033  ‘cfv 6328  t+ctcl 14340 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-cleq 2794 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator