Theorem 2exsb 1351

Description: An equivalent expression for double existence.

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

Proof of Theorem 2exsb

Step	Hyp	Ref	Expression
1		exsb 1350	. . . 4 |- (E.yph <-> E.wA.y(y = w -> ph))
2	1	exbii 1051	. . 3 |- (E.xE.yph <-> E.xE.wA.y(y = w -> ph))
3		excom 1046	. . 3 |- (E.xE.wA.y(y = w -> ph) <-> E.wE.xA.y(y = w -> ph))
4	2, 3	bitr 173	. 2 |- (E.xE.yph <-> E.wE.xA.y(y = w -> ph))
5		exsb 1350	. . . 4 |- (E.xA.y(y = w -> ph) <-> E.zA.x(x = z -> A.y(y = w -> ph)))
6		impexp 347	. . . . . . . 8 |- (((x = z /\ y = w) -> ph) <-> (x = z -> (y = w -> ph)))
7	6	albii 999	. . . . . . 7 |- (A.y((x = z /\ y = w) -> ph) <-> A.y(x = z -> (y = w -> ph)))
8		19.21v 1285	. . . . . . 7 |- (A.y(x = z -> (y = w -> ph)) <-> (x = z -> A.y(y = w -> ph)))
9	7, 8	bitr2 174	. . . . . 6 |- ((x = z -> A.y(y = w -> ph)) <-> A.y((x = z /\ y = w) -> ph))
10	9	albii 999	. . . . 5 |- (A.x(x = z -> A.y(y = w -> ph)) <-> A.xA.y((x = z /\ y = w) -> ph))
11	10	exbii 1051	. . . 4 |- (E.zA.x(x = z -> A.y(y = w -> ph)) <-> E.zA.xA.y((x = z /\ y = w) -> ph))
12	5, 11	bitr 173	. . 3 |- (E.xA.y(y = w -> ph) <-> E.zA.xA.y((x = z /\ y = w) -> ph))
13	12	exbii 1051	. 2 |- (E.wE.xA.y(y = w -> ph) <-> E.wE.zA.xA.y((x = z /\ y = w) -> ph))
14		excom 1046	. 2 |- (E.wE.zA.xA.y((x = z /\ y = w) -> ph) <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))
15	4, 13, 14	3bitr 177	1 |- (E.xE.yph <-> E.zE.wA.xA.y((x = z /\ y = w) -> ph))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980

This theorem is referenced by:  2eu6 1454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172

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