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Theorem 2eu7 1977
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 1366 . . . 4 (xφxxφ)
21hbeu 1904 . . 3 (∃!yxφx∃!yxφ)
32euan 1939 . 2 (∃!x(∃!yxφ yφ) ↔ (∃!yxφ ∃!xyφ))
4 ancom 253 . . . . 5 ((xφ yφ) ↔ (yφ xφ))
54eubii 1892 . . . 4 (∃!y(xφ yφ) ↔ ∃!y(yφ xφ))
6 hbe1 1366 . . . . 5 (yφyyφ)
76euan 1939 . . . 4 (∃!y(yφ xφ) ↔ (yφ ∃!yxφ))
8 ancom 253 . . . 4 ((yφ ∃!yxφ) ↔ (∃!yxφ yφ))
95, 7, 83bitri 195 . . 3 (∃!y(xφ yφ) ↔ (∃!yxφ yφ))
109eubii 1892 . 2 (∃!x∃!y(xφ yφ) ↔ ∃!x(∃!yxφ yφ))
11 ancom 253 . 2 ((∃!xyφ ∃!yxφ) ↔ (∃!yxφ ∃!xyφ))
123, 10, 113bitr4ri 202 1 ((∃!xyφ ∃!yxφ) ↔ ∃!x∃!y(xφ yφ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1363  ∃!weu 1883
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-eu 1886  df-mo 1887
This theorem is referenced by: (None)
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