Theorem ralabso

Description: Simplification of restricted quantification in a transitive class. When ph is quantifier-free, this shows that the formula A. x e. y ph is absolute for transitive models, which is a particular case of Lemma I.16.2 of Kunen2 p. 95. (Contributed by Eric Schmidt, 19-Oct-2025)

	Ref	Expression
Assertion	ralabso	|- ( ( Tr M /\ A e. M ) -> ( A. x e. A ph <-> A. x e. M ( x e. A -> ph ) ) )

Proof

Step	Hyp	Ref	Expression
1		trss	|- ( Tr M -> ( A e. M -> A C_ M ) )
2	1	imp	|- ( ( Tr M /\ A e. M ) -> A C_ M )
3		ralss	|- ( A C_ M -> ( A. x e. A ph <-> A. x e. M ( x e. A -> ph ) ) )
4	2 3	syl	|- ( ( Tr M /\ A e. M ) -> ( A. x e. A ph <-> A. x e. M ( x e. A -> ph ) ) )