Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege64b Structured version   Visualization version   GIF version

Theorem frege64b 43233
Description: Lemma for frege65b 43234. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege64b (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))

Proof of Theorem frege64b
StepHypRef Expression
1 frege62b 43231 . 2 ([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒))
2 frege18 43142 . 2 (([𝑧 / 𝑦]𝜓 → (∀𝑦(𝜓𝜒) → [𝑧 / 𝑦]𝜒)) → (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒))))
31, 2ax-mp 5 1 (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1531  [wsb 2059
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-10 2129  ax-12 2163  ax-frege1 43114  ax-frege2 43115  ax-frege8 43133  ax-frege58b 43225
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-ex 1774  df-nf 1778  df-sb 2060
This theorem is referenced by:  frege65b  43234
  Copyright terms: Public domain W3C validator