Theorem ralbiim 2566

Description: Split a biconditional and distribute quantifier. (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 385	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	ralbii 2441	. 2 |- ( A. x e. A ( ph <-> ps ) <-> A. x e. A ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		r19.26 2558	. 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 183	1 |- ( A. x e. A ( ph <-> ps ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wral 2416

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 1423  ax-gen 1425  ax-4 1487  ax-17 1506

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-ral 2421

This theorem is referenced by:  eqreu  2876