Theorem ralnexOLD 2993

Description: Obsolete proof of ralnex 2992 as of 16-Jul-2021. (Contributed by NM, 21-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem ralnexOLD

Step	Hyp	Ref	Expression
1		df-ral 2917	. 2 |- ( A. x e. A -. ph <-> A. x ( x e. A -> -. ph ) )
2		alinexa 1770	. . 3 |- ( A. x ( x e. A -> -. ph ) <-> -. E. x ( x e. A /\ ph ) )
3		df-rex 2918	. . 3 |- ( E. x e. A ph <-> E. x ( x e. A /\ ph ) )
4	2, 3	xchbinxr 325	. 2 |- ( A. x ( x e. A -> -. ph ) <-> -. E. x e. A ph )
5	1, 4	bitri 264	1 |- ( A. x e. A -. ph <-> -. E. x e. A ph )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ral 2917  df-rex 2918

This theorem is referenced by: (None)