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Theorem 2eu7 1994
 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 1384 . . . 4 (∃𝑥𝜑 → ∀𝑥𝑥𝜑)
21hbeu 1921 . . 3 (∃!𝑦𝑥𝜑 → ∀𝑥∃!𝑦𝑥𝜑)
32euan 1956 . 2 (∃!𝑥(∃!𝑦𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃!𝑦𝑥𝜑 ∧ ∃!𝑥𝑦𝜑))
4 ancom 253 . . . . 5 ((∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ (∃𝑦𝜑 ∧ ∃𝑥𝜑))
54eubii 1909 . . . 4 (∃!𝑦(∃𝑥𝜑 ∧ ∃𝑦𝜑) ↔ ∃!𝑦(∃𝑦𝜑 ∧ ∃𝑥𝜑))
6 hbe1 1384 . . . . 5 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
76euan 1956 . . . 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 1909 . 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 1381  ∃!weu 1900 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904 This theorem is referenced by: (None)
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