Theorem r19.21t 2394

Description: Theorem 19.21 of [Margaris] p. 90 with restricted quantifiers (closed theorem version). (Contributed by NM, 1-Mar-2008.)

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		bi2.04 237	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
2	1	albii 1359	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
3		19.21t 1474	. . 3 |- ( F/ x ph -> ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
4	2, 3	syl5bb 181	. 2 |- ( F/ x ph -> ( A. x ( x e. A -> ( ph -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
5		df-ral 2311	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		df-ral 2311	. . 3 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7	6	imbi2i 215	. 2 |- ( ( ph -> A. x e. A ps ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
8	4, 5, 7	3bitr4g 212	1 |- ( F/ x ph -> ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 98   A.wal 1241   F/wnf 1349   e. wcel 1393   A.wral 2306

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-4 1400  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311

This theorem is referenced by:  r19.21  2395