Metamath Proof Explorer

Theorem frege73

Description: Lemma for frege87 . Proposition 73 of Frege1879 p. 59. (Contributed by RP, 28-Mar-2020) (Revised by RP, 5-Jul-2020) (Proof modification is discouraged.)

	Ref	Expression
Hypotheses	frege73.x	$⊢ X \in U$
	frege73.y	$⊢ Y \in V$
Assertion	frege73	$⊢ (R hereditary A \to X \in A) \to (R hereditary A \to (X R Y \to Y \in A))$

Proof

Step	Hyp	Ref	Expression
1		frege73.x	$⊢ X \in U$
2		frege73.y	$⊢ Y \in V$
3	1 2	frege72	$⊢ R hereditary A \to (X \in A \to (X R Y \to Y \in A))$
4		ax-frege2	$⊢ (R hereditary A \to (X \in A \to (X R Y \to Y \in A))) \to ((R hereditary A \to X \in A) \to (R hereditary A \to (X R Y \to Y \in A)))$
5	3 4	ax-mp	$⊢ (R hereditary A \to X \in A) \to (R hereditary A \to (X R Y \to Y \in A))$