Metamath Proof Explorer

Theorem frege64a

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

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

Proof

Step Hyp Ref Expression
1 frege62a 
 |-  ( if- ( si , ch , et ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> if- ( si , th , ze ) ) )
2 frege18 
 |-  ( ( 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 
 |-  ( ( if- ( ph , ps , ta ) -> if- ( si , ch , et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( si , th , ze ) ) ) )