Theorem ralbi 2750

Description: Distribute a restricted universal quantifier over a biconditional. Theorem 19.15 of [Margaris] p. 90 with restricted quantification. (Contributed by NM, 6-Oct-2003.)

Assertion

Ref	Expression
ralbi	|- (A.x e. A (ph <-> ps) -> (A.x e. A ph <-> A.x e. A ps))

Proof of Theorem ralbi

Step	Hyp	Ref	Expression
1		nfra1 2664	. 2 |- F/xA.x e. A (ph <-> ps)
2		rsp 2674	. . 3 |- (A.x e. A (ph <-> ps) -> (x e. A -> (ph <-> ps)))
3	2	imp 418	. 2 |- ((A.x e. A (ph <-> ps) /\ x e. A) -> (ph <-> ps))
4	1, 3	ralbida 2628	1 |- (A.x e. A (ph <-> ps) -> (A.x e. A ph <-> A.x e. A ps))

Syntax hints:   -> wi 4   <-> wb 176   e. wcel 1710  A.wral 2614

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-6 1729  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2619

This theorem is referenced by:  uniiunlem  3353  iineq2  3986