Theorem frege59b 39037

Description: A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 38946 incorrectly referenced where frege30 38965 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege59b	⊢ ([𝑥 / 𝑦]𝜑 → (¬ [𝑥 / 𝑦]𝜓 → ¬ ∀𝑦(𝜑 → 𝜓)))

Proof of Theorem frege59b

Step	Hyp	Ref	Expression
1		frege58bcor 39036	. 2 ⊢ (∀𝑦(𝜑 → 𝜓) → ([𝑥 / 𝑦]𝜑 → [𝑥 / 𝑦]𝜓))
2		frege30 38965	. 2 ⊢ ((∀𝑦(𝜑 → 𝜓) → ([𝑥 / 𝑦]𝜑 → [𝑥 / 𝑦]𝜓)) → ([𝑥 / 𝑦]𝜑 → (¬ [𝑥 / 𝑦]𝜓 → ¬ ∀𝑦(𝜑 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ([𝑥 / 𝑦]𝜑 → (¬ [𝑥 / 𝑦]𝜓 → ¬ ∀𝑦(𝜑 → 𝜓)))

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1654  [wsb 2067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220  ax-13 2389  ax-frege1 38923  ax-frege2 38924  ax-frege8 38942  ax-frege28 38963  ax-frege58b 39034

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068

This theorem is referenced by: (None)