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Theorem 2eu2ex 1989
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 1930 . 2 (∃!𝑥∃!𝑦𝜑 → ∃𝑥∃!𝑦𝜑)
2 euex 1930 . . 3 (∃!𝑦𝜑 → ∃𝑦𝜑)
32eximi 1491 . 2 (∃𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)
41, 3syl 14 1 (∃!𝑥∃!𝑦𝜑 → ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  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-nf 1350  df-sb 1646  df-eu 1903
This theorem is referenced by: (None)
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