Theorem ralbiim 3069

Description: Split a biconditional and distribute quantifier. Restricted quantifier version of albiim 1816. (Contributed by NM, 3-Jun-2012.)

Assertion

Ref	Expression
ralbiim	|- ( A. x e. A ( ph <-> ps ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )

Proof of Theorem ralbiim

Step	Hyp	Ref	Expression
1		dfbi2 660	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	ralbii 2980	. 2 |- ( A. x e. A ( ph <-> ps ) <-> A. x e. A ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		r19.26 3064	. 2 |- ( A. x e. A ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )
4	2, 3	bitri 264	1 |- ( A. x e. A ( ph <-> ps ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   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

This theorem depends on definitions:  df-bi 197  df-an 386  df-ral 2917

This theorem is referenced by:  eqreu  3398  isclo2  20892  chrelat4i  29232  hlateq  34685  ntrneik13  38396  ntrneix13  38397  2ralbiim  41174