Theorem r19.21t

Description: Restricted quantifier version of 19.21t ; closed form of r19.21 . (Contributed by NM, 1-Mar-2008) (Proof shortened by Wolf Lammen, 2-Jan-2020)

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

Proof

Step	Hyp	Ref	Expression
1		19.21t	|- ( F/ x ph -> ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
2		df-ral	|- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3		bi2.04	|- ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
4	3	albii	|- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
5	2 4	bitri	|- ( A. x e. A ( ph -> ps ) <-> A. x ( ph -> ( x e. A -> 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	1 5 7	3bitr4g	|- ( F/ x ph -> ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) )