Theorem r19.28zf

Description: Restricted quantifier version of Theorem 19.28 of Margaris p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	r19.28zf.1	|- F/ x ph
	r19.28zf.2	|- F/_ x A
Assertion	r19.28zf	|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )

Proof

Step	Hyp	Ref	Expression
1		r19.28zf.1	|- F/ x ph
2		r19.28zf.2	|- F/_ x A
3		r19.26	|- ( A. x e. A ( ph /\ ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) )
4	1 2	r19.3rzf	|- ( A =/= (/) -> ( ph <-> A. x e. A ph ) )
5	4	anbi1d	|- ( A =/= (/) -> ( ( ph /\ A. x e. A ps ) <-> ( A. x e. A ph /\ A. x e. A ps ) ) )
6	3 5	bitr4id	|- ( A =/= (/) -> ( A. x e. A ( ph /\ ps ) <-> ( ph /\ A. x e. A ps ) ) )