Theorem exanali 1585

Description: A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)

Assertion

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

Proof of Theorem exanali

Step	Hyp	Ref	Expression
1		annim 414	. . 3 |- ((ph /\ -. ps) <-> -. (ph -> ps))
2	1	exbii 1582	. 2 |- (E.x(ph /\ -. ps) <-> E.x -. (ph -> ps))
3		exnal 1574	. 2 |- (E.x -. (ph -> ps) <-> -. A.x(ph -> ps))
4	2, 3	bitri 240	1 |- (E.x(ph /\ -. ps) <-> -. A.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-1 5  ax-2 6  ax-3 7  ax-mp 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:  ax11indn  2195  rexnal  2625  gencbval  2903  nss  3329  ssfin  4470  ncfinlowerlem1  4482  spfinex  4537  nfunv  5138  funsex  5828  fnfullfunlem1  5856  foundex  5914  fnfreclem1  6315