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Theorem 2euex 2035
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 1995 . 2 (∃!𝑥𝑦𝜑 ↔ (∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑))
2 excom 1599 . . . 4 (∃𝑥𝑦𝜑 ↔ ∃𝑦𝑥𝜑)
3 hbe1 1429 . . . . . 6 (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
43hbmo 1987 . . . . 5 (∃*𝑥𝑦𝜑 → ∀𝑦∃*𝑥𝑦𝜑)
5 19.8a 1527 . . . . . . 7 (𝜑 → ∃𝑦𝜑)
65moimi 2013 . . . . . 6 (∃*𝑥𝑦𝜑 → ∃*𝑥𝜑)
7 df-mo 1952 . . . . . 6 (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑))
86, 7sylib 120 . . . . 5 (∃*𝑥𝑦𝜑 → (∃𝑥𝜑 → ∃!𝑥𝜑))
94, 8eximdh 1547 . . . 4 (∃*𝑥𝑦𝜑 → (∃𝑦𝑥𝜑 → ∃𝑦∃!𝑥𝜑))
102, 9syl5bi 150 . . 3 (∃*𝑥𝑦𝜑 → (∃𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑))
1110impcom 123 . 2 ((∃𝑥𝑦𝜑 ∧ ∃*𝑥𝑦𝜑) → ∃𝑦∃!𝑥𝜑)
121, 11sylbi 119 1 (∃!𝑥𝑦𝜑 → ∃𝑦∃!𝑥𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wex 1426  ∃!weu 1948  ∃*wmo 1949
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by: (None)
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