Theorem r19.21v

Description: Restricted quantifier version of 19.21v . (Contributed by NM, 15-Oct-2003) (Proof shortened by Andrew Salmon, 30-May-2011) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020)

	Ref	Expression
Assertion	r19.21v	|- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )

Proof

Step	Hyp	Ref	Expression
1		bi2.04	|- ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
2	1	albii	|- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
3		19.21v	|- ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
4	2 3	bitri	|- ( A. x ( x e. A -> ( ph -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
5		df-ral	|- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		df-ral	|- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7	6	imbi2i	|- ( ( ph -> A. x e. A ps ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
8	4 5 7	3bitr4i	|- ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) )