Theorem r19.35 2758

Description: Restricted quantifier version of Theorem 19.35 of [Margaris] p. 90. (Contributed by NM, 20-Sep-2003.)

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 2746	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> (A.x e. A ph /\ A.x e. A -. ps))
2		annim 414	. . . . 5 |- ((ph /\ -. ps) <-> -. (ph -> ps))
3	2	ralbii 2638	. . . 4 |- (A.x e. A (ph /\ -. ps) <-> A.x e. A -. (ph -> ps))
4		df-an 360	. . . 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	3bitr3i 266	. . 3 |- (A.x e. A -. (ph -> ps) <-> -. (A.x e. A ph -> -. A.x e. A -. ps))
6	5	con2bii 322	. 2 |- ((A.x e. A ph -> -. A.x e. A -. ps) <-> -. A.x e. A -. (ph -> ps))
7		dfrex2 2627	. . 3 |- (E.x e. A ps <-> -. A.x e. A -. ps)
8	7	imbi2i 303	. 2 |- ((A.x e. A ph -> E.x e. A ps) <-> (A.x e. A ph -> -. A.x e. A -. ps))
9		dfrex2 2627	. 2 |- (E.x e. A (ph -> ps) <-> -. A.x e. A -. (ph -> ps))
10	6, 8, 9	3bitr4ri 269	1 |- (E.x e. A (ph -> ps) <-> (A.x e. A ph -> E.x e. A ps))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wral 2614  E.wrex 2615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-ral 2619  df-rex 2620

This theorem is referenced by:  r19.36av  2759  r19.37  2760  r19.43  2766  r19.37zv  3646  r19.36zv  3650