Theorem ralnex 2359

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 2354	. 2 |- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
2		alinexa 1535	. . 3 |- ( A. x ( x e. A -> -. ph ) <-> -. E. x ( x e. A /\ ph ) )
3		df-rex 2355	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4	2, 3	xchbinxr 641	. 2 |- ( A. x ( x e. A -> -. ph ) <-> -. E. x e. A ph )
5	1, 4	bitri 182	1 |- ( A. x e. A -. ph <-> -. E. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   A.wal 1283   E.wex 1422   e. wcel 1434   A.wral 2349   E.wrex 2350

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-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie2 1424

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-ral 2354  df-rex 2355

This theorem is referenced by:  rexalim  2362  ralinexa  2394  nrex  2454  nrexdv  2455  uni0b  3634  iindif2m  3753  supmoti  6465  suprnubex  8098  icc0r  9025  ioo0  9346  ico0  9348  ioc0  9349  prmind2  10646  sqrt2irr  10685