Theorem albiim 1498

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 388	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	albii 1481	. 2 |- ( A. x ( ph <-> ps ) <-> A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		19.26 1492	. 2 |- ( A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
4	2, 3	bitri 184	1 |- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   A.wal 1362

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 1458  ax-gen 1460

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  2albiim  1499  hbbid  1586  equveli  1770  spsbbi  1855  eu1  2067  eqss  3194  ssext  4250