Theorem r19.21t 2541

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		bi2.04 247	. . . 4 |- ( ( x e. A -> ( ph -> ps ) ) <-> ( ph -> ( x e. A -> ps ) ) )
2	1	albii 1458	. . 3 |- ( A. x ( x e. A -> ( ph -> ps ) ) <-> A. x ( ph -> ( x e. A -> ps ) ) )
3		19.21t 1570	. . 3 |- ( F/ x ph -> ( A. x ( ph -> ( x e. A -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
4	2, 3	syl5bb 191	. 2 |- ( F/ x ph -> ( A. x ( x e. A -> ( ph -> ps ) ) <-> ( ph -> A. x ( x e. A -> ps ) ) ) )
5		df-ral 2449	. 2 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
6		df-ral 2449	. . 3 |- ( A. x e. A ps <-> A. x ( x e. A -> ps ) )
7	6	imbi2i 225	. 2 |- ( ( ph -> A. x e. A ps ) <-> ( ph -> A. x ( x e. A -> ps ) ) )
8	4, 5, 7	3bitr4g 222	1 |- ( F/ x ph -> ( A. x e. A ( ph -> ps ) <-> ( ph -> A. x e. A ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   A.wal 1341   F/wnf 1448   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449

This theorem is referenced by:  r19.21  2542