Theorem frege113

Description: Proposition 113 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	frege112.z	|- Z e. V
Assertion	frege113	|- ( ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> Z = X ) ) -> ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege112.z	|- Z e. V
2	1	frege112	|- ( Z = X -> X ( ( t+ ` R ) u. _I ) Z )
3		frege7	|- ( ( Z = X -> X ( ( t+ ` R ) u. _I ) Z ) -> ( ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> Z = X ) ) -> ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) ) ) )
4	2 3	ax-mp	|- ( ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> Z = X ) ) -> ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) ) )