Theorem ralbi 2598

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 2497	. 2 |- F/ x A. x e. A ( ph <-> ps )
2		rsp 2513	. . 3 |- ( A. x e. A ( ph <-> ps ) -> ( x e. A -> ( ph <-> ps ) ) )
3	2	imp 123	. 2 |- ( ( A. x e. A ( ph <-> ps ) /\ x e. A ) -> ( ph <-> ps ) )
4	1, 3	ralbida 2460	1 |- ( A. x e. A ( ph <-> ps ) -> ( A. x e. A ph <-> A. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 104   e. wcel 2136   A.wral 2444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-gen 1437  ax-4 1498  ax-ial 1522

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-ral 2449

This theorem is referenced by:  uniiunlem  3231  iineq2  3883  ralrnmpt  5627  f1mpt  5739  mpo2eqb  5951  ralrnmpo  5956  cau3lem  11056