Metamath Proof Explorer


Theorem frege86

Description: Conclusion about element one past Y in the R -sequence. Proposition 86 of Frege1879 p. 66. (Contributed by RP, 1-Jul-2020) (Revised by RP, 7-Jul-2020) (Proof modification is discouraged.)

Ref Expression
Hypotheses frege86.x
|- X e. U
frege86.y
|- Y e. V
frege86.r
|- R e. W
frege86.a
|- A e. B
Assertion 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 ) ) ) ) )

Proof

Step Hyp Ref Expression
1 frege86.x
 |-  X e. U
2 frege86.y
 |-  Y e. V
3 frege86.r
 |-  R e. W
4 frege86.a
 |-  A e. B
5 1 2 3 4 frege85
 |-  ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> Y e. A ) ) )
6 frege19
 |-  ( ( X ( t+ ` R ) Y -> ( A. w ( X R w -> w e. A ) -> ( R hereditary A -> Y e. A ) ) ) -> ( ( ( 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 ) ) ) ) ) )
7 5 6 ax-mp
 |-  ( ( ( 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 ) ) ) ) )