Metamath Proof Explorer

Theorem frege66a

Description: Swap antecedents of frege65a . Proposition 66 of Frege1879 p. 54. (Contributed by RP, 17-Apr-2020) (Proof modification is discouraged.)

Ref Expression
Assertion frege66a 
|- ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )

Proof

Step Hyp Ref Expression
1 frege65a 
 |-  ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )
2 ax-frege8 
 |-  ( ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) ) -> ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) ) )
3 1 2 ax-mp 
 |-  ( ( ( ch -> th ) /\ ( et -> ze ) ) -> ( ( ( ps -> ch ) /\ ( ta -> et ) ) -> ( if- ( ph , ps , ta ) -> if- ( ph , th , ze ) ) ) )