Theorem ralbiim 2441

Description: Split a biconditional and distribute quantifier. (Contributed by NM, 3-Jun-2012.)

Assertion

Ref	Expression
ralbiim	|- (A.x e. A (ph <-> ps) <-> (A.x e. A (ph -> ps) /\ A.x e. A (ps -> ph)))

Proof of Theorem ralbiim

Step	Hyp	Ref	Expression
1		dfbi2 368	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
2	1	ralbii 2324	. 2 |- (A.x e. A (ph <-> ps) <-> A.x e. A ((ph -> ps) /\ (ps -> ph)))
3		r19.26 2435	. 2 |- (A.x e. A ((ph -> ps) /\ (ps -> ph)) <-> (A.x e. A (ph -> ps) /\ A.x e. A (ps -> ph)))
4	2, 3	bitri 173	1 |- (A.x e. A (ph <-> ps) <-> (A.x e. A (ph -> ps) /\ A.x e. A (ps -> ph)))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98  A.wral 2300

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-gen 1335  ax-4 1397  ax-17 1416

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-ral 2305

This theorem is referenced by:  eqreu  2727