Theorem exanali 1043

Description: A transformation of quantifiers and logical connectives.

Assertion

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

Proof of Theorem exanali

Step	Hyp	Ref	Expression
1		iman 237	. . . 4 |- ((ph -> ps) <-> -. (ph /\ -. ps))
2	1	albii 999	. . 3 |- (A.x(ph -> ps) <-> A.x -. (ph /\ -. ps))
3		alnex 1033	. . 3 |- (A.x -. (ph /\ -. ps) <-> -. E.x(ph /\ -. ps))
4	2, 3	bitr 173	. 2 |- (A.x(ph -> ps) <-> -. E.x(ph /\ -. ps))
5	4	con2bii 221	1 |- (E.x(ph /\ -. ps) <-> -. A.x(ph -> ps))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954  E.wex 980

This theorem is referenced by:  ax11indn 1366  rexnal 1654  gencbval 1840  prlem934 5139  reclem2pr 5157

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-4 973  ax-5o 975

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981

Copyright terms: Public domain