Theorem albiim 1611

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 609	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
2	1	albii 1566	. 2 |- (A.x(ph <-> ps) <-> A.x((ph -> ps) /\ (ps -> ph)))
3		19.26 1593	. 2 |- (A.x((ph -> ps) /\ (ps -> ph)) <-> (A.x(ph -> ps) /\ A.x(ps -> ph)))
4	2, 3	bitri 240	1 |- (A.x(ph <-> ps) <-> (A.x(ph -> ps) /\ A.x(ps -> ph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360

This theorem is referenced by:  2albiim  1612  equveli  1988  eu1  2225  eqss  3288