Metamath Proof Explorer

Theorem frege88

Description: Commuted form of frege87 . Proposition 88 of Frege1879 p. 67. (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 frege88 ⊢ Y R Z → X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → 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 1 2 3 4 5 frege87 ⊢ X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Y R Z → Z ∈ A
7 frege15 ⊢ X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Y R Z → Z ∈ A → Y R Z → X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Z ∈ A
8 6 7 ax-mp ⊢ Y R Z → X t+ ⁡ R Y → ∀ w X R w → w ∈ A → R hereditary A → Z ∈ A