Theorem albiim 1762

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 638	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	albii 1701	. 2 |- ( A. x ( ph <-> ps ) <-> A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		19.26 1742	. 2 |- ( A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
4	2, 3	bitri 257	1 |- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   A.wal 1452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692

This theorem depends on definitions:  df-bi 190  df-an 377

This theorem is referenced by:  2albiim  1763  mo2v  2316  eu1  2349  eqss  3458  ssext  4668  asymref2  5235  pm14.122a  36816