Theorem ralbida

Description: Formula-building rule for restricted universal quantifier (deduction form). (Contributed by NM, 6-Oct-2003)

	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	pm5.74da	|- ( ph -> ( ( x e. A -> ps ) <-> ( x e. A -> ch ) ) )
4	1 3	albid	|- ( ph -> ( A. x ( x e. A -> ps ) <-> A. x ( x e. A -> ch ) ) )
5		df-ral	|- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
6		df-ral	|- ( A. x e. A ch <-> A. x ( x e. A -> ch ) )
7	4 5 6	3bitr4g	|- ( ph -> ( A. x e. A ps <-> A. x e. A ch ) )