Theorem alinexa 1578

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)

Assertion

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

Proof of Theorem alinexa

Step	Hyp	Ref	Expression
1		imnan 411	. . 3 |- ((ph -> -. ps) <-> -. (ph /\ ps))
2	1	albii 1566	. 2 |- (A.x(ph -> -. ps) <-> A.x -. (ph /\ ps))
3		alnex 1543	. 2 |- (A.x -. (ph /\ ps) <-> -. E.x(ph /\ ps))
4	2, 3	bitri 240	1 |- (A.x(ph -> -. ps) <-> -. E.x(ph /\ ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  equs3  1644  axi11e  2332  ralnex  2625