Theorem frege65a 38177

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege65a	|- ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )

Proof of Theorem frege65a

Step	Hyp	Ref	Expression
1		ifpimim 37854	. . 3 |- (if- ( ph , ( ps -> ch ) , ( ta -> et ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , ch , et ) ) )
2		frege64a 38176	. . 3 |- ( (if- ( ph , ps , ta ) -> if- ( ph , ch , et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )
3	1, 2	syl 17	. 2 |- (if- ( ph , ( ps -> ch ) , ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )
4		frege61a 38173	. 2 |- ( (if- ( ph , ( ps -> ch ) , ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) ) -> ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) ) )
5	3, 4	ax-mp 5	1 |- ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )

Syntax hints:   -> 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-frege58a 38169

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

This theorem is referenced by:  frege66a  38178