Theorem rexim 2719

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		con3 126	. . . 4 |- ((ph -> ps) -> (-. ps -> -. ph))
2	1	ral2imi 2691	. . 3 |- (A.x e. A (ph -> ps) -> (A.x e. A -. ps -> A.x e. A -. ph))
3	2	con3d 125	. 2 |- (A.x e. A (ph -> ps) -> (-. A.x e. A -. ph -> -. A.x e. A -. ps))
4		dfrex2 2628	. 2 |- (E.x e. A ph <-> -. A.x e. A -. ph)
5		dfrex2 2628	. 2 |- (E.x e. A ps <-> -. A.x e. A -. ps)
6	3, 4, 5	3imtr4g 261	1 |- (A.x e. A (ph -> ps) -> (E.x e. A ph -> E.x e. A ps))

Syntax hints:  -. wn 3   -> wi 4  A.wral 2615  E.wrex 2616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-ral 2620  df-rex 2621

This theorem is referenced by:  reximia  2720  reximdai  2723  r19.29  2755  reupick2  3542  ss2iun  3985