Theorem frege114

Description: If X belongs to the R -sequence beginning with Z , then Z belongs to the R -sequence beginning with X or X follows Z in the R -sequence. Proposition 114 of Frege1879 p. 76. (Contributed by RP, 7-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege114.x	|- X e. U
	frege114.z	|- Z e. V
Assertion	frege114	|- ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) )

Proof

Step	Hyp	Ref	Expression
1		frege114.x	|- X e. U
2		frege114.z	|- Z e. V
3	1	frege104	|- ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> Z = X ) )
4	2	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 ) ) )
5	3 4	ax-mp	|- ( Z ( ( t+ ` R ) u. _I ) X -> ( -. Z ( t+ ` R ) X -> X ( ( t+ ` R ) u. _I ) Z ) )