Theorem 2exsb 2132

Description: An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)

Assertion

Ref	Expression
2exsb	|- (E.xE.yph <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))

Distinct variable groups:   x,y,z   y,w,z   ph,z,w

Allowed substitution hints:   ph(x,y)

Proof of Theorem 2exsb

Step	Hyp	Ref	Expression
1		exsb 2130	. . . 4 |- (E.yph <-> E.wA.y(y = w -> ph))
2	1	exbii 1582	. . 3 |- (E.xE.yph <-> E.xE.wA.y(y = w -> ph))
3		excom 1741	. . 3 |- (E.xE.wA.y(y = w -> ph) <-> E.wE.xA.y(y = w -> ph))
4	2, 3	bitri 240	. 2 |- (E.xE.yph <-> E.wE.xA.y(y = w -> ph))
5		exsb 2130	. . . . 5 |- (E.xA.y(y = w -> ph) <-> E.zA.x(x = z -> A.y(y = w -> ph)))
6		impexp 433	. . . . . . . . 9 |- (((x = z /\ y = w) -> ph) <-> (x = z -> (y = w -> ph)))
7	6	albii 1566	. . . . . . . 8 |- (A.y((x = z /\ y = w) -> ph) <-> A.y(x = z -> (y = w -> ph)))
8		19.21v 1890	. . . . . . . 8 |- (A.y(x = z -> (y = w -> ph)) <-> (x = z -> A.y(y = w -> ph)))
9	7, 8	bitr2i 241	. . . . . . 7 |- ((x = z -> A.y(y = w -> ph)) <-> A.y((x = z /\ y = w) -> ph))
10	9	albii 1566	. . . . . 6 |- (A.x(x = z -> A.y(y = w -> ph)) <-> A.xA.y((x = z /\ y = w) -> ph))
11	10	exbii 1582	. . . . 5 |- (E.zA.x(x = z -> A.y(y = w -> ph)) <-> E.zA.xA.y((x = z /\ y = w) -> ph))
12	5, 11	bitri 240	. . . 4 |- (E.xA.y(y = w -> ph) <-> E.zA.xA.y((x = z /\ y = w) -> ph))
13	12	exbii 1582	. . 3 |- (E.wE.xA.y(y = w -> ph) <-> E.wE.zA.xA.y((x = z /\ y = w) -> ph))
14		excom 1741	. . 3 |- (E.wE.zA.xA.y((x = z /\ y = w) -> ph) <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))
15	13, 14	bitri 240	. 2 |- (E.wE.xA.y(y = w -> ph) <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))
16	4, 15	bitri 240	1 |- (E.xE.yph <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  2eu6  2289