Theorem albiim 1475

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 386	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	albii 1458	. 2 |- ( A. x ( ph <-> ps ) <-> A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		19.26 1469	. 2 |- ( A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
4	2, 3	bitri 183	1 |- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   A.wal 1341

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 1435  ax-gen 1437

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  2albiim  1476  hbbid  1563  equveli  1747  spsbbi  1832  eu1  2039  eqss  3157  ssext  4199