Theorem frege67a 38179

Description: Lemma for frege68a 38180. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege67a	|- ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) )

Proof of Theorem frege67a

Step	Hyp	Ref	Expression
1		ax-frege58a 38169	. 2 |- ( ( ps /\ ch ) -> if- ( ph , ps , ch ) )
2		frege7 38102	. 2 |- ( ( ( ps /\ ch ) -> if- ( ph , ps , ch ) ) -> ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ( ( ps /\ ch ) <-> th ) -> ( th -> ( ps /\ ch ) ) ) -> ( ( ( ps /\ ch ) <-> th ) -> ( th -> if- ( ph , ps , ch ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-frege1 38084  ax-frege2 38085  ax-frege58a 38169

This theorem is referenced by:  frege68a  38180