Theorem ralbi 2638

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 2537	. 2 |- F/ x A. x e. A ( ph <-> ps )
2		rsp 2553	. . 3 |- ( A. x e. A ( ph <-> ps ) -> ( x e. A -> ( ph <-> ps ) ) )
3	2	imp 124	. 2 |- ( ( A. x e. A ( ph <-> ps ) /\ x e. A ) -> ( ph <-> ps ) )
4	1, 3	ralbida 2500	1 |- ( A. x e. A ( ph <-> ps ) -> ( A. x e. A ph <-> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2176   A.wral 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1470  ax-gen 1472  ax-4 1533  ax-ial 1557

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-ral 2489

This theorem is referenced by:  uniiunlem  3282  iineq2  3944  ralrnmpt  5722  f1mpt  5840  mpo2eqb  6055  ralrnmpo  6060  cau3lem  11425