Theorem frege111

Description: If Y belongs to the R -sequence beginning with Z , then every result of an application of the procedure R to Y belongs to the R -sequence beginning with Z or precedes Z in the R -sequence. Proposition 111 of Frege1879 p. 75. (Contributed by RP, 7-Jul-2020) (Revised by RP, 8-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege111.z	|- Z e. A
	frege111.y	|- Y e. B
	frege111.v	|- V e. C
	frege111.r	|- R e. D
Assertion	frege111	|- ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> ( -. V ( t+ ` R ) Z -> Z ( ( t+ ` R ) u. _I ) V ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege111.z	|- Z e. A
2		frege111.y	|- Y e. B
3		frege111.v	|- V e. C
4		frege111.r	|- R e. D
5	1 2 3 4	frege108	|- ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> Z ( ( t+ ` R ) u. _I ) V ) )
6		frege25	|- ( ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> Z ( ( t+ ` R ) u. _I ) V ) ) -> ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> ( -. V ( t+ ` R ) Z -> Z ( ( t+ ` R ) u. _I ) V ) ) ) )
7	5 6	ax-mp	|- ( Z ( ( t+ ` R ) u. _I ) Y -> ( Y R V -> ( -. V ( t+ ` R ) Z -> Z ( ( t+ ` R ) u. _I ) V ) ) )