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Theorem 2eu7 2039
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 1427 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbeu 1966 . . 3 (∃!𝑦𝑥𝜑 → ∀𝑥∃!𝑦𝑥𝜑)
32euan 2001 . 2 (∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
4 ancom 262 . . . . 5 ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
54eubii 1954 . . . 4 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6 hbe1 1427 . . . . 5 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
76euan 2001 . . . 4 (∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑) ↔ (∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑))
8 ancom 262 . . . 4 ((∃𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
95, 7, 83bitri 204 . . 3 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
109eubii 1954 . 2 (∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑))
11 ancom 262 . 2 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
123, 10, 113bitr4ri 211 1 ((∃!𝑥𝑦𝜑 ∧ ∃!𝑦𝑥𝜑) ↔ ∃!𝑥∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103  wex 1424  ∃!weu 1945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949
This theorem is referenced by: (None)
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