Theorem 2eu7 1432

Description: Two equivalent expressions for double existential uniqueness.

Assertion

Ref	Expression
2eu7	|- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))

Proof of Theorem 2eu7

Step	Hyp	Ref	Expression
1		hbe1 990	. . . 4 |- (E.xph -> A.xE.xph)
2	1	hbeu 1366	. . 3 |- (E!yE.xph -> A.xE!yE.xph)
3	2	euan 1405	. 2 |- (E!x(E!yE.xph /\ E.yph) <-> (E!yE.xph /\ E!xE.yph))
4		ancom 435	. . . . 5 |- ((E.xph /\ E.yph) <-> (E.yph /\ E.xph))
5	4	eubii 1364	. . . 4 |- (E!y(E.xph /\ E.yph) <-> E!y(E.yph /\ E.xph))
6		hbe1 990	. . . . 5 |- (E.yph -> A.yE.yph)
7	6	euan 1405	. . . 4 |- (E!y(E.yph /\ E.xph) <-> (E.yph /\ E!yE.xph))
8		ancom 435	. . . 4 |- ((E.yph /\ E!yE.xph) <-> (E!yE.xph /\ E.yph))
9	5, 7, 8	3bitr 177	. . 3 |- (E!y(E.xph /\ E.yph) <-> (E!yE.xph /\ E.yph))
10	9	eubii 1364	. 2 |- (E!xE!y(E.xph /\ E.yph) <-> E!x(E!yE.xph /\ E.yph))
11		ancom 435	. 2 |- ((E!xE.yph /\ E!yE.xph) <-> (E!yE.xph /\ E!xE.yph))
12	3, 10, 11	3bitr4r 184	1 |- ((E!xE.yph /\ E!yE.xph) <-> E!xE!y(E.xph /\ E.yph))

Syntax hints:   <-> wb 146   /\ wa 223  E.wex 956  E!weu 1357

This theorem is referenced by:  2eu8 1433

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360

Copyright terms: Public domain