Theorem ralinexa 2577

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.)

Assertion

Ref	Expression
ralinexa	|- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) )

Proof of Theorem ralinexa

Step	Hyp	Ref	Expression
1		imnan 701	. . 3 |- ( ( ph -> -. ps ) <-> -. ( ph /\ ps ) )
2	1	ralbii 2556	. 2 |- ( A. x e. A ( ph -> -. ps ) <-> A. x e. A -. ( ph /\ ps ) )
3		ralnex 2538	. 2 |- ( A. x e. A -. ( ph /\ ps ) <-> -. E. x e. A ( ph /\ ps ) )
4	2, 3	bitri 184	1 |- ( A. x e. A ( ph -> -. ps ) <-> -. E. x e. A ( ph /\ ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   A.wral 2528   E.wrex 2529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-5 1500  ax-gen 1502  ax-ie2 1547  ax-4 1563  ax-17 1579

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-fal 1408  df-nf 1514  df-ral 2533  df-rex 2534

This theorem is referenced by:  ntreq0  15159