Theorem exanali 725

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 205	. . . 4 |- ((ph -> ps) <-> -. (ph /\ -. ps))
2	1	bial 695	. . 3 |- (A.x(ph -> ps) <-> A.x -. (ph /\ -. ps))
3		alnex 716	. . 3 |- (A.x -. (ph /\ -. ps) <-> -. E.x(ph /\ -. ps))
4	2, 3	bitr 151	. 2 |- (A.x(ph -> ps) <-> -. E.x(ph /\ -. ps))
5	4	bicon2i 194	1 |- (E.x(ph /\ -. ps) <-> -. A.x(ph -> ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wal 672  E.wex 678

This theorem is referenced by:  rexnal 1210  gencbval 1373  prlem934 3933  reclem2pr 3951

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org