Theorem ralinexa 2660

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 411	. . 3 |- ((ph -> -. ps) <-> -. (ph /\ ps))
2	1	ralbii 2639	. 2 |- (A.x e. A (ph -> -. ps) <-> A.x e. A -. (ph /\ ps))
3		ralnex 2625	. 2 |- (A.x e. A -. (ph /\ ps) <-> -. E.x e. A (ph /\ ps))
4	2, 3	bitri 240	1 |- (A.x e. A (ph -> -. ps) <-> -. E.x e. A (ph /\ ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  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  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2620  df-rex 2621

This theorem is referenced by: (None)