Theorem ralbiim 2447

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 368	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	ralbii 2330	. 2 |- ( A. x e. A ( ph <-> ps ) <-> A. x e. A ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		r19.26 2441	. 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 173	1 |- ( A. x e. A ( ph <-> ps ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   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-17 1419

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-ral 2311

This theorem is referenced by:  eqreu  2733