Theorem 2reu5lem1 3042

Description: Lemma for 2reu5 3045. Note that E!x e. AE!y e. Bph does not mean "there is exactly one x in A and exactly one y in B such that ph holds;" see comment for 2eu5 2288. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Assertion

Ref	Expression
2reu5lem1	|- (E!x e. A E!y e. B ph <-> E!xE!y(x e. A /\ y e. B /\ ph))

Distinct variable groups:   y,A   x,B   x,y

Allowed substitution hints:   ph(x,y)   A(x)   B(y)

Proof of Theorem 2reu5lem1

Step	Hyp	Ref	Expression
1		df-reu 2622	. . 3 |- (E!y e. B ph <-> E!y(y e. B /\ ph))
2	1	reubii 2798	. 2 |- (E!x e. A E!y e. B ph <-> E!x e. A E!y(y e. B /\ ph))
3		df-reu 2622	. . 3 |- (E!x e. A E!y(y e. B /\ ph) <-> E!x(x e. A /\ E!y(y e. B /\ ph)))
4		euanv 2265	. . . . . 6 |- (E!y(x e. A /\ (y e. B /\ ph)) <-> (x e. A /\ E!y(y e. B /\ ph)))
5	4	bicomi 193	. . . . 5 |- ((x e. A /\ E!y(y e. B /\ ph)) <-> E!y(x e. A /\ (y e. B /\ ph)))
6		3anass 938	. . . . . . 7 |- ((x e. A /\ y e. B /\ ph) <-> (x e. A /\ (y e. B /\ ph)))
7	6	bicomi 193	. . . . . 6 |- ((x e. A /\ (y e. B /\ ph)) <-> (x e. A /\ y e. B /\ ph))
8	7	eubii 2213	. . . . 5 |- (E!y(x e. A /\ (y e. B /\ ph)) <-> E!y(x e. A /\ y e. B /\ ph))
9	5, 8	bitri 240	. . . 4 |- ((x e. A /\ E!y(y e. B /\ ph)) <-> E!y(x e. A /\ y e. B /\ ph))
10	9	eubii 2213	. . 3 |- (E!x(x e. A /\ E!y(y e. B /\ ph)) <-> E!xE!y(x e. A /\ y e. B /\ ph))
11	3, 10	bitri 240	. 2 |- (E!x e. A E!y(y e. B /\ ph) <-> E!xE!y(x e. A /\ y e. B /\ ph))
12	2, 11	bitri 240	1 |- (E!x e. A E!y e. B ph <-> E!xE!y(x e. A /\ y e. B /\ ph))

Syntax hints:   <-> wb 176   /\ wa 358   /\ w3a 934   e. wcel 1710  E!weu 2204  E!wreu 2617

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-reu 2622

This theorem is referenced by:  2reu5lem3  3044