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 ∈ U
frege87.y ⊢ Y ∈ V
frege87.z ⊢ Z ∈ W
frege87.r ⊢ R ∈ S
frege87.a ⊢ A ∈ B
Assertion frege87 ⊢ X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Y R Z → Z ∈ A

Proof

Step Hyp Ref Expression
1 frege87.x ⊢ X ∈ U
2 frege87.y ⊢ Y ∈ V
3 frege87.z ⊢ Z ∈ W
4 frege87.r ⊢ R ∈ S
5 frege87.a ⊢ A ∈ B
6 2 3 frege73 ⊢ R hereditary A → Y ∈ A → R hereditary A → Y R Z → Z ∈ A
7 1 2 4 5 frege86 ⊢ R hereditary A → Y ∈ A → R hereditary A → Y R Z → Z ∈ A → X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Y R Z → Z ∈ A
8 6 7 ax-mp ⊢ X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Y R Z → Z ∈ A