Theorem rexbi 2638

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 2536	. 2 |- F/ x A. x e. A ( ph <-> ps )
2		rsp 2552	. . 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	rexbida 2500	1 |- ( A. x e. A ( ph <-> ps ) -> ( E. x e. A ph <-> E. x e. A ps ) )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2175   A.wral 2483   E.wrex 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 1469  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-4 1532  ax-ial 1556

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-ral 2488  df-rex 2489

This theorem is referenced by:  rexrnmpo  6060