Theorem albiim 1666

Description: Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)

Assertion

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

Proof of Theorem albiim

Step	Hyp	Ref	Expression
1		dfbi2 628	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	albii 1611	. 2 |- ( A. x ( ph <-> ps ) <-> A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		19.26 1648	. 2 |- ( A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
4	2, 3	bitri 249	1 |- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   A.wal 1368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603

This theorem depends on definitions:  df-bi 185  df-an 371

This theorem is referenced by:  2albiim  1667  mo2v  2269  mo2vOLD  2270  mo2vOLDOLD  2271  eu1  2312  eu1OLD  2313  eqss  3480  ssext  4656  asymref2  5324  pm14.122a  29825