Theorem rexnalim 2311

Description: Relationship between restricted universal and existential quantifiers. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
rexnalim	|- (E.x e. A -. ph -> -. A.x e. A ph)

Proof of Theorem rexnalim

Step	Hyp	Ref	Expression
1		df-rex 2306	. 2 |- (E.x e. A -. ph <-> E.x(x e. A /\ -. ph))
2		exanaliim 1535	. . 3 |- (E.x(x e. A /\ -. ph) -> -. A.x(x e. A -> ph))
3		df-ral 2305	. . 3 |- (A.x e. A ph <-> A.x(x e. A -> ph))
4	2, 3	sylnibr 601	. 2 |- (E.x(x e. A /\ -. ph) -> -. A.x e. A ph)
5	1, 4	sylbi 114	1 |- (E.x e. A -. ph -> -. A.x e. A ph)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 97  A.wal 1240  E.wex 1378   e. wcel 1390  A.wral 2300  E.wrex 2301

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-4 1397  ax-17 1416  ax-ial 1424

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-ral 2305  df-rex 2306

This theorem is referenced by:  ralexim  2312  iundif2ss  3713