Theorem r19.21t 2955

Description: Restricted quantifier version of 19.21t 2073; closed form of r19.21 2956. (Contributed by NM, 1-Mar-2008.) (Proof shortened by Wolf Lammen, 2-Jan-2020.)

Assertion

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

Proof of Theorem r19.21t

Step	Hyp	Ref	Expression
1		19.21t 2073	. 2 |- ( F/ x ph -> ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
2		df-ral 2917	. . 3 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
3		bi2.04 376	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
4	3	albii 1747	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
5	2, 4	bitri 264	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
6		df-ral 2917	. . 3 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7	6	imbi2i 326	. 2 |- ( ( ph -> A. x e. A ps ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
8	1, 5, 7	3bitr4g 303	1 |- ( F/ x ph -> ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   F/wnf 1708   e. wcel 1990   A.wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710  df-ral 2917

This theorem is referenced by:  r19.21  2956