Theorem frege81

Description: If X has a property A that is hereditary in the R -sequence, and if Y follows X in the R -sequence, then Y has property A . This is a form of induction attributed to Jakob Bernoulli. Proposition 81 of Frege1879 p. 63. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege81.x	|- X e. U
	frege81.y	|- Y e. V
	frege81.r	|- R e. W
	frege81.a	|- A e. B
Assertion	frege81	|- ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege81.x	|- X e. U
2		frege81.y	|- Y e. V
3		frege81.r	|- R e. W
4		frege81.a	|- A e. B
5		vex	|- a e. _V
6	1 5	frege74	|- ( X e. A -> ( R hereditary A -> ( X R a -> a e. A ) ) )
7	6	alrimdv	|- ( X e. A -> ( R hereditary A -> A. a ( X R a -> a e. A ) ) )
8	1 2 3 4	frege80	|- ( ( X e. A -> ( R hereditary A -> A. a ( X R a -> a e. A ) ) ) -> ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )
9	7 8	ax-mp	|- ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )