Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege67c Structured version   Visualization version   GIF version

Theorem frege67c 43597
Description: Lemma for frege68c 43598. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege67c (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))

Proof of Theorem frege67c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege58c 43588 . 2 (∀𝑥𝜑[𝐴 / 𝑥]𝜑)
3 frege7 43475 . 2 ((∀𝑥𝜑[𝐴 / 𝑥]𝜑) → (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑))))
42, 3ax-mp 5 1 (((∀𝑥𝜑𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑𝜓) → (𝜓[𝐴 / 𝑥]𝜑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1532  wcel 2099  [wsbc 3776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-frege1 43457  ax-frege2 43458  ax-frege58b 43568
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-sbc 3777
This theorem is referenced by:  frege68c  43598
  Copyright terms: Public domain W3C validator