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Theorem 2eu7 1924
Description: Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
Assertion
Ref Expression
2eu7 ((∃!xyφ ∃!yxφ) ↔ ∃!x∃!y(xφ yφ))

Proof of Theorem 2eu7
StepHypRef Expression
1 hbe1 1331 . . . 4 (xφxxφ)
21hbeu 1852 . . 3 (∃!yxφx∃!yxφ)
32euan 1886 . 2 (∃!x(∃!yxφ yφ) ↔ (∃!yxφ ∃!xyφ))
4 ancom 251 . . . . 5 ((xφ yφ) ↔ (yφ xφ))
54eubii 1842 . . . 4 (∃!y(xφ yφ) ↔ ∃!y(yφ xφ))
6 hbe1 1331 . . . . 5 (yφyyφ)
76euan 1886 . . . 4 (∃!y(yφ xφ) ↔ (yφ ∃!yxφ))
8 ancom 251 . . . 4 ((yφ ∃!yxφ) ↔ (∃!yxφ yφ))
95, 7, 83bitri 193 . . 3 (∃!y(xφ yφ) ↔ (∃!yxφ yφ))
109eubii 1842 . 2 (∃!x∃!y(xφ yφ) ↔ ∃!x(∃!yxφ yφ))
11 ancom 251 . 2 ((∃!xyφ ∃!yxφ) ↔ (∃!yxφ ∃!xyφ))
123, 10, 113bitr4ri 200 1 ((∃!xyφ ∃!yxφ) ↔ ∃!x∃!y(xφ yφ))
Colors of variables: wff set class
Syntax hints:   wa 95  wb 96  wex 1328  ∃!weu 1833
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 97  ax-ia2 98  ax-ia3 99  ax-io 612  ax-5 1282  ax-7 1283  ax-gen 1284  ax-ie1 1329  ax-ie2 1330  ax-8 1344  ax-10 1345  ax-11 1346  ax-i12 1347  ax-bnd 1348  ax-4 1349  ax-17 1367  ax-i9 1371  ax-ial 1376  ax-i5r 1377
This theorem depends on definitions:  df-bi 108  df-tru 1204  df-nf 1296  df-sb 1586  df-eu 1836  df-mo 1837
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