Theorem frege82

Description: Closed-form deduction based on frege81 . Proposition 82 of Frege1879 p. 64. (Contributed by RP, 1-Jul-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege82.x	|- X e. U
	frege82.y	|- Y e. V
	frege82.r	|- R e. W
	frege82.a	|- A e. B
Assertion	frege82	|- ( ( ph -> X e. A ) -> ( R hereditary A -> ( ph -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		frege82.x	|- X e. U
2		frege82.y	|- Y e. V
3		frege82.r	|- R e. W
4		frege82.a	|- A e. B
5	1 2 3 4	frege81	|- ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) )
6		frege18	|- ( ( X e. A -> ( R hereditary A -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) -> ( ( ph -> X e. A ) -> ( R hereditary A -> ( ph -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) ) )
7	5 6	ax-mp	|- ( ( ph -> X e. A ) -> ( R hereditary A -> ( ph -> ( X ( t+ ` R ) Y -> Y e. A ) ) ) )