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 | ||
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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 ) ) ) ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | frege86.x | |- X e. U |
|
2 | frege86.y | |- Y e. V |
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3 | frege86.r | |- R e. W |
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4 | frege86.a | |- A e. B |
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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 ) ) ) ) ) |