Theorem ralbida

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003) (Proof shortened by Wolf Lammen, 31-Oct-2024)

	Ref	Expression
Hypotheses	ralbida.1	|- F/ x ph
	ralbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
Assertion	ralbida	|- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )

Proof

Step	Hyp	Ref	Expression
1		ralbida.1	|- F/ x ph
2		ralbida.2	|- ( ( ph /\ x e. A ) -> ( ps <-> ch ) )
3	2	biimpd	|- ( ( ph /\ x e. A ) -> ( ps -> ch ) )
4	1 3	ralimdaa	|- ( ph -> ( A. x e. A ps -> A. x e. A ch ) )
5	2	biimprd	|- ( ( ph /\ x e. A ) -> ( ch -> ps ) )
6	1 5	ralimdaa	|- ( ph -> ( A. x e. A ch -> A. x e. A ps ) )
7	4 6	impbid	|- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )