Theorem frege59b 44658

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 44567 incorrectly referenced where frege30 44586 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

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

Proof of Theorem frege59b

Step	Hyp	Ref	Expression
1		frege58bcor 44657	. 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
2		frege30 44586	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) → ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567  [wsb 2095

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212  ax-frege1 44544  ax-frege2 44545  ax-frege8 44563  ax-frege28 44584  ax-frege58b 44655

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-nf 1813  df-sb 2096

This theorem is used by: (None)