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Theorem 2euex 1987
 Description: Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
2euex (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)

Proof of Theorem 2euex
StepHypRef Expression
1 eu5 1947 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 excom 1554 . . . 4 (∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
3 hbe1 1384 . . . . . 6 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
43hbmo 1939 . . . . 5 (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝑦𝜑)
5 19.8a 1482 . . . . . . 7 (𝜑 → ∃𝑦𝜑)
65moimi 1965 . . . . . 6 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
7 df-mo 1904 . . . . . 6 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
86, 7sylib 127 . . . . 5 (∃*𝑥𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
94, 8eximdh 1502 . . . 4 (∃*𝑥𝑦𝜑 → (∃𝑦𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
102, 9syl5bi 141 . . 3 (∃*𝑥𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
1110impcom 116 . 2 ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
121, 11sylbi 114 1 (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97  ∃wex 1381  ∃!weu 1900  ∃*wmo 1901 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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