Theorem ralbi 2683

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 2581	. 2 |- F/ x A. x e. A ( ph <-> ps )
2		rsp 2597	. . 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 2544	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 2209   A.wral 2528

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 1500  ax-gen 1502  ax-4 1563  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-ral 2533

This theorem is referenced by:  uniiunlem  3338  iineq2  4024  ralrnmpt  5841  f1mpt  5967  mpo2eqb  6188  ralrnmpo  6193  cau3lem  11858