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Theorem 2eu2ex 2066
Description: Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
Assertion
Ref Expression
2eu2ex (∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)

Proof of Theorem 2eu2ex
StepHypRef Expression
1 euex 2007 . 2 (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃!𝑦𝜑)
2 euex 2007 . . 3 (∃!𝑦𝜑 → ∃𝑦𝜑)
32eximi 1564 . 2 (∃𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)
41, 3syl 14 1 (∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wex 1453  ∃!weu 1977
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980
This theorem is referenced by: (None)
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