Theorem ralnex 2625

Description: Relationship between restricted universal and existential quantifiers. (Contributed by NM, 21-Jan-1997.)

Assertion

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

Proof of Theorem ralnex

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 |- (A.x e. A -. ph <-> A.x(x e. A -> -. ph))
2		alinexa 1578	. . 3 |- (A.x(x e. A -> -. ph) <-> -. E.x(x e. A /\ ph))
3		df-rex 2621	. . 3 |- (E.x e. A ph <-> E.x(x e. A /\ ph))
4	2, 3	xchbinxr 302	. 2 |- (A.x(x e. A -> -. ph) <-> -. E.x e. A ph)
5	1, 4	bitri 240	1 |- (A.x e. A -. ph <-> -. E.x e. A ph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   e. wcel 1710  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:  dfrex2  2628  ralinexa  2660  nrex  2717  nrexdv  2718  r19.43  2767  rabeq0  3573  iindif2  4036  evenodddisj  4517  rexiunxp  4825