Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege61c Structured version   Visualization version   GIF version

Theorem frege61c 39059
Description: Lemma for frege65c 39063. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege61c (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓))

Proof of Theorem frege61c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege58c 39056 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege9 38947 . 2 ((∀𝑥𝜑[𝐴 / 𝑥]𝜑) → (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓)))
42, 3ax-mp 5 1 (([𝐴 / 𝑥]𝜑𝜓) → (∀𝑥𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1656  wcel 2166  [wsbc 3663
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2804  ax-frege1 38925  ax-frege2 38926  ax-frege8 38944  ax-frege58b 39036
This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2813  df-cleq 2819  df-clel 2822  df-v 3417  df-sbc 3664
This theorem is referenced by:  frege65c  39063
  Copyright terms: Public domain W3C validator