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Theorem eupickb 2119
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eupickb ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2117 . . 3 ((∃!𝑥𝜑 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
213adant2 1018 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
3 3simpc 998 . . 3 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)))
4 pm3.22 265 . . . . 5 ((𝜑𝜓) → (𝜓𝜑))
54eximi 1611 . . . 4 (∃𝑥(𝜑𝜓) → ∃𝑥(𝜓𝜑))
65anim2i 342 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)))
7 eupick 2117 . . 3 ((∃!𝑥𝜓 ∧ ∃𝑥(𝜓𝜑)) → (𝜓𝜑))
83, 6, 73syl 17 . 2 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜓𝜑))
92, 8impbid 129 1 ((∃!𝑥𝜑 ∧ ∃!𝑥𝜓 ∧ ∃𝑥(𝜑𝜓)) → (𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980  wex 1503  ∃!weu 2038
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546
This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042
This theorem is referenced by: (None)
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