Theorem frege64a 38176

Description: Lemma for frege65a 38177. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege64a	|- ( (if- ( ph , ps , ta ) -> if- ( si , ch , et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( si , th , ze ) ) ) )

Proof of Theorem frege64a

Step	Hyp	Ref	Expression
1		frege62a 38174	. 2 |- (if- ( si , ch , et ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> if- ( si , th , ze ) ) )
2		frege18 38112	. 2 |- ( (if- ( si , ch , et ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> if- ( si , th , ze ) ) ) -> ( (if- ( ph , ps , ta ) -> if- ( si , ch , et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( si , th , ze ) ) ) ) )
3	1, 2	ax-mp 5	1 |- ( (if- ( ph , ps , ta ) -> if- ( si , ch , et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> (if- ( ph , ps , ta ) -> if- ( si , 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:  frege65a  38177