Theorem frege72

Description: If property A is hereditary in the R -sequence, if x has property A , and if y is a result of an application of the procedure R to x , then y has property A . Proposition 72 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege72.x	|- X e. U
	frege72.y	|- Y e. V
Assertion	frege72	|- ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege72.x	|- X e. U
2		frege72.y	|- Y e. V
3	2	frege58c	|- ( A. z ( X R z -> z e. A ) -> [. Y / z ]. ( X R z -> z e. A ) )
4		sbcim1	|- ( [. Y / z ]. ( X R z -> z e. A ) -> ( [. Y / z ]. X R z -> [. Y / z ]. z e. A ) )
5		sbcbr2g	|- ( Y e. V -> ( [. Y / z ]. X R z <-> X R [_ Y / z ]_ z ) )
6		csbvarg	|- ( Y e. V -> [_ Y / z ]_ z = Y )
7	6	breq2d	|- ( Y e. V -> ( X R [_ Y / z ]_ z <-> X R Y ) )
8	5 7	bitrd	|- ( Y e. V -> ( [. Y / z ]. X R z <-> X R Y ) )
9	2 8	ax-mp	|- ( [. Y / z ]. X R z <-> X R Y )
10		sbcel1v	|- ( [. Y / z ]. z e. A <-> Y e. A )
11	4 9 10	3imtr3g	|- ( [. Y / z ]. ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) )
12	3 11	syl	|- ( A. z ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) )
13	1	frege71	|- ( ( A. z ( X R z -> z e. A ) -> ( X R Y -> Y e. A ) ) -> ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) ) )
14	12 13	ax-mp	|- ( R hereditary A -> ( X e. A -> ( X R Y -> Y e. A ) ) )