Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege63c Structured version   Visualization version   GIF version

Theorem frege63c 43250
Description: Analogue of frege63b 43232. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
frege59c.a 𝐴𝐵
Assertion
Ref Expression
frege63c ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))

Proof of Theorem frege63c
StepHypRef Expression
1 frege59c.a . . 3 𝐴𝐵
21frege62c 43249 . 2 ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))
3 frege24 43139 . 2 (([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)) → ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒))))
42, 3ax-mp 5 1 ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑𝜒) → [𝐴 / 𝑥]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  wcel 2098  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege58b 43225
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-sbc 3773
This theorem is referenced by:  frege91  43278
  Copyright terms: Public domain W3C validator