Theorem ralbiim 2631

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 388	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	ralbii 2503	. 2 |- ( A. x e. A ( ph <-> ps ) <-> A. x e. A ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		r19.26 2623	. 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 184	1 |- ( A. x e. A ( ph <-> ps ) <-> ( A. x e. A ( ph -> ps ) /\ A. x e. A ( ps -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wral 2475

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1461  ax-gen 1463  ax-4 1524  ax-17 1540

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-ral 2480

This theorem is referenced by:  eqreu  2956