Metamath Proof Explorer


Theorem frege87

Description: If Z is a result of an application of the procedure R to an object Y that follows X in the R -sequence and if every result of an application of the procedure R to X has a property A that is hereditary in the R -sequence, then Z has property A . Proposition 87 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege87.x
|- X e. U
frege87.y
|- Y e. V
frege87.z
|- Z e. W
frege87.r
|- R e. S
frege87.a
|- A e. B
Assertion frege87
|- ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) )

Proof

Step Hyp Ref Expression
1 frege87.x
 |-  X e. U
2 frege87.y
 |-  Y e. V
3 frege87.z
 |-  Z e. W
4 frege87.r
 |-  R e. S
5 frege87.a
 |-  A e. B
6 2 3 frege73
 |-  ( ( R hereditary A -> Y e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) )
7 1 2 4 5 frege86
 |-  ( ( ( R hereditary A -> Y e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) -> ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) ) )
8 6 7 ax-mp
 |-  ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> ( Y R Z -> Z e. A ) ) ) )