Theorem rexim 2564

Description: Theorem 19.22 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 22-Nov-1994.) (Proof shortened by Andrew Salmon, 30-May-2011.)

Assertion

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

Proof of Theorem rexim

Step	Hyp	Ref	Expression
1		df-ral 2453	. . . 4 |- ( A. x e. A ( ph -> ps ) <-> A. x ( x e. A -> ( ph -> ps ) ) )
2		simpl 108	. . . . . . 7 |- ( ( x e. A /\ ph ) -> x e. A )
3	2	a1i 9	. . . . . 6 |- ( ( x e. A -> ( ph -> ps ) ) -> ( ( x e. A /\ ph ) -> x e. A ) )
4		pm3.31 260	. . . . . 6 |- ( ( x e. A -> ( ph -> ps ) ) -> ( ( x e. A /\ ph ) -> ps ) )
5	3, 4	jcad 305	. . . . 5 |- ( ( x e. A -> ( ph -> ps ) ) -> ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
6	5	alimi 1448	. . . 4 |- ( A. x ( x e. A -> ( ph -> ps ) ) -> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
7	1, 6	sylbi 120	. . 3 |- ( A. x e. A ( ph -> ps ) -> A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) )
8		exim 1592	. . 3 |- ( A. x ( ( x e. A /\ ph ) -> ( x e. A /\ ps ) ) -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. A /\ ps ) ) )
9	7, 8	syl 14	. 2 |- ( A. x e. A ( ph -> ps ) -> ( E. x ( x e. A /\ ph ) -> E. x ( x e. A /\ ps ) ) )
10		df-rex 2454	. 2 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
11		df-rex 2454	. 2 |- ( E. x e. A ps <-> E. x ( x e. A /\ ps ) )
12	9, 10, 11	3imtr4g 204	1 |- ( A. x e. A ( ph -> ps ) -> ( E. x e. A ph -> E. x e. A ps ) )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1346   E.wex 1485   e. wcel 2141   A.wral 2448   E.wrex 2449

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 1440  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-ral 2453  df-rex 2454

This theorem is referenced by:  reximia  2565  reximdai  2568  r19.29  2607  reupick2  3413  ss2iun  3888  chfnrn  5607