Theorem r19.35 1298

Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90.

Assertion

Ref	Expression
r19.35	|- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))

Proof of Theorem r19.35

Step	Hyp	Ref	Expression
1		r19.26 1289	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> (A.x e. A ph /\ A.x e. A -. ps))
2		annim 206	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	biral 1223	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> A.x e. A -. (ph -> ps))
4		df-an 198	. . . 4 |- ((A.x e. A ph /\ A.x e. A -. ps) <-> -. (A.x e. A ph -> -. A.x e. A -. ps))
5	1, 3, 4	3bitr3 156	. . 3 |- (A.x e. A -. (ph -> ps) <-> -. (A.x e. A ph -> -. A.x e. A -. ps))
6	5	bicon2i 194	. 2 |- ((A.x e. A ph -> -. A.x e. A -. ps) <-> -. A.x e. A -. (ph -> ps))
7		dfrex2 1212	. . 3 |- (E.x e. A ps <-> -. A.x e. A -. ps)
8	7	imbi2i 160	. 2 |- ((A.x e. A ph -> E.x e. A ps) <-> (A.x e. A ph -> -. A.x e. A -. ps))
9		dfrex2 1212	. 2 |- (E.x e. A (ph -> ps) <-> -. A.x e. A -. (ph -> ps))
10	6, 8, 9	3bitr4r 159	1 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   /\ wa 196  A.wral 1201  E.wrex 1202

This theorem is referenced by:  r19.36av 1299  r19.37av 1300  r19.36zv 1772

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-or 197  df-an 198  df-ex 679  df-ral 1205  df-rex 1206

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