Theorem albiim 1746

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 632	. . 3 |- ( ( ph <-> ps ) <-> ( ( ph -> ps ) /\ ( ps -> ph ) ) )
2	1	albii 1685	. 2 |- ( A. x ( ph <-> ps ) <-> A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) )
3		19.26 1726	. 2 |- ( A. x ( ( ph -> ps ) /\ ( ps -> ph ) ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )
4	2, 3	bitri 252	1 |- ( A. x ( ph <-> ps ) <-> ( A. x ( ph -> ps ) /\ A. x ( ps -> ph ) ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   A.wal 1435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676

This theorem depends on definitions:  df-bi 188  df-an 372

This theorem is referenced by:  2albiim  1747  mo2v  2283  eu1  2316  eqss  3422  ssext  4619  asymref2  5179  pm14.122a  36686