Theorem frege102

Description: If Z belongs to the R -sequence beginning with X , then every result of an application of the procedure R to Z follows X in the R -sequence. Proposition 102 of Frege1879 p. 72. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege102.x	|- X e. A
	frege102.z	|- Z e. B
	frege102.v	|- V e. C
	frege102.r	|- R e. D
Assertion	frege102	|- ( X ( ( t+ ` R ) u. _I ) Z -> ( Z R V -> X ( t+ ` R ) V ) )

Proof

Step	Hyp	Ref	Expression
1		frege102.x	|- X e. A
2		frege102.z	|- Z e. B
3		frege102.v	|- V e. C
4		frege102.r	|- R e. D
5	2 3 4	frege92	|- ( Z = X -> ( Z R V -> X ( t+ ` R ) V ) )
6	1 2 3 4	frege96	|- ( X ( t+ ` R ) Z -> ( Z R V -> X ( t+ ` R ) V ) )
7	2	frege101	|- ( ( Z = X -> ( Z R V -> X ( t+ ` R ) V ) ) -> ( ( X ( t+ ` R ) Z -> ( Z R V -> X ( t+ ` R ) V ) ) -> ( X ( ( t+ ` R ) u. _I ) Z -> ( Z R V -> X ( t+ ` R ) V ) ) ) )
8	5 6 7	mp2	|- ( X ( ( t+ ` R ) u. _I ) Z -> ( Z R V -> X ( t+ ` R ) V ) )