Theorem ralbiim 2752

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 609	. . 3 |- ((ph <-> ps) <-> ((ph -> ps) /\ (ps -> ph)))
2	1	ralbii 2639	. 2 |- (A.x e. A (ph <-> ps) <-> A.x e. A ((ph -> ps) /\ (ps -> ph)))
3		r19.26 2747	. 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 240	1 |- (A.x e. A (ph <-> ps) <-> (A.x e. A (ph -> ps) /\ A.x e. A (ps -> ph)))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wral 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  eqreu  3029  ssofeq  4078