Theorem frege59a 38171

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 38107 incorrectly referenced where frege30 38126 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege59a	|- (if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) )

Proof of Theorem frege59a

Step	Hyp	Ref	Expression
1		frege58acor 38170	. 2 |- ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> (if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) )
2		frege30 38126	. 2 |- ( ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> (if- ( ph , ps , th ) -> if- ( ph , ch , ta ) ) ) -> (if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) ) )
3	1, 2	ax-mp 5	1 |- (if- ( ph , ps , th ) -> ( -. if- ( ph , ch , ta ) -> -. ( ( ps -> ch ) /\ ( th -> ta ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege28 38124  ax-frege58a 38169

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by: (None)