Theorem rexbi 2490

Description: Distribute a restricted existential quantifier over a biconditional. Theorem 19.18 of [Margaris] p. 90 with restricted quantification. (Contributed by Jim Kingdon, 21-Jan-2019.)

Assertion

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

Proof of Theorem rexbi

Step	Hyp	Ref	Expression
1		nfra1 2397	. 2 |- F/ x A. x e. A ( ph <-> ps )
2		rsp 2411	. . 3 |- ( A. x e. A ( ph <-> ps ) -> ( x e. A -> ( ph <-> ps ) ) )
3	2	imp 122	. 2 |- ( ( A. x e. A ( ph <-> ps ) /\ x e. A ) -> ( ph <-> ps ) )
4	1, 3	rexbida 2363	1 |- ( A. x e. A ( ph <-> ps ) -> ( E. x e. A ph <-> E. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 103   e. wcel 1433   A.wral 2348   E.wrex 2349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-4 1440  ax-ial 1467

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-ral 2353  df-rex 2354

This theorem is referenced by:  rexrnmpt2  5636