Theorem ralnex 2316

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 2311	. 2 |- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
2		alinexa 1494	. . 3 |- ( A. x ( x e. A -> -. ph ) <-> -. E. x ( x e. A /\ ph ) )
3		df-rex 2312	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4	2, 3	xchbinxr 608	. 2 |- ( A. x ( x e. A -> -. ph ) <-> -. E. x e. A ph )
5	1, 4	bitri 173	1 |- ( A. x e. A -. ph <-> -. E. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97   <-> wb 98   A.wal 1241   E.wex 1381   e. wcel 1393   A.wral 2306   E.wrex 2307

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-gen 1338  ax-ie2 1383

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-ral 2311  df-rex 2312

This theorem is referenced by:  rexalim  2319  ralinexa  2351  nrex  2411  nrexdv  2412  uni0b  3605  iindif2m  3724  icc0r  8795  sqrt2irr  9878