Theorem frege125

Description: Lemma for frege126 . Proposition 125 of Frege1879 p. 81. (Contributed by RP, 9-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege123.x	|- X e. U
	frege123.y	|- Y e. V
	frege124.m	|- M e. W
	frege124.r	|- R e. S
Assertion	frege125	|- ( ( X ( ( t+ ` R ) u. _I ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege123.x	|- X e. U
2		frege123.y	|- Y e. V
3		frege124.m	|- M e. W
4		frege124.r	|- R e. S
5	1 2 3 4	frege124	|- ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) )
6		frege20	|- ( ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> X ( ( t+ ` R ) u. _I ) M ) ) ) -> ( ( X ( ( t+ ` R ) u. _I ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) ) )
7	5 6	ax-mp	|- ( ( X ( ( t+ ` R ) u. _I ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) -> ( Fun `' `' R -> ( Y R X -> ( Y ( t+ ` R ) M -> ( -. X ( t+ ` R ) M -> M ( ( t+ ` R ) u. _I ) X ) ) ) ) )