Theorem frege101

Description: Lemma for frege102 . Proposition 101 of Frege1879 p. 72. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege99.z	|- Z e. U
Assertion	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 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege99.z	|- Z e. U
2	1	frege100	|- ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> Z = X ) )
3		frege48	|- ( ( X ( ( t+ ` R ) u. _I ) Z -> ( -. X ( t+ ` R ) Z -> Z = X ) ) -> ( ( 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 ) ) ) ) )
4	2 3	ax-mp	|- ( ( 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 ) ) ) )