Theorem rexanaliim 2614

Description: A transformation of restricted quantifiers and logical connectives. (Contributed by NM, 4-Sep-2005.) (Revised by Jim Kingdon, 18-Jan-2026.)

Assertion

Ref	Expression
rexanaliim	|- ( E. x e. A ( ph /\ -. ps ) -> -. A. x e. A ( ph -> ps ) )

Proof of Theorem rexanaliim

Step	Hyp	Ref	Expression
1		annimim 688	. . 3 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )
2	1	reximi 2605	. 2 |- ( E. x e. A ( ph /\ -. ps ) -> E. x e. A -. ( ph -> ps ) )
3		rexnalim 2497	. 2 |- ( E. x e. A -. ( ph -> ps ) -> -. A. x e. A ( ph -> ps ) )
4	2, 3	syl 14	1 |- ( E. x e. A ( ph /\ -. ps ) -> -. A. x e. A ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   A.wral 2486   E.wrex 2487

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-4 1534  ax-17 1550  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-ral 2491  df-rex 2492

This theorem is referenced by:  umgr2edg1  15964