Metamath Proof Explorer

Theorem frege63a

Description: Proposition 63 of Frege1879 p. 52. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

Ref Expression
Assertion frege63a 
|- ( if- ( ph , ps , th ) -> ( et -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) ) )

Proof

Step Hyp Ref Expression
1 frege62a 
 |-  ( if- ( ph , ps , th ) -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) )
2 frege24 
 |-  ( ( if- ( ph , ps , th ) -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) ) -> ( if- ( ph , ps , th ) -> ( et -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) ) ) )
3 1 2 ax-mp 
 |-  ( if- ( ph , ps , th ) -> ( et -> ( ( ( ps -> ch ) /\ ( th -> ta ) ) -> if- ( ph , ch , ta ) ) ) )