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Theorem reupick 3281
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reupick (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem reupick
StepHypRef Expression
1 ssel 3017 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21ad2antrr 472 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
3 df-rex 2365 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2366 . . . . . 6 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
53, 4anbi12i 448 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑)))
61ancrd 319 . . . . . . . . . . 11 (𝐴𝐵 → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
76anim1d 329 . . . . . . . . . 10 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
8 an32 529 . . . . . . . . . 10 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∧ 𝑥𝐴))
97, 8syl6ib 159 . . . . . . . . 9 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
109eximdv 1808 . . . . . . . 8 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
11 eupick 2027 . . . . . . . . 9 ((∃!𝑥(𝑥𝐵𝜑) ∧ ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1211ex 113 . . . . . . . 8 (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴) → ((𝑥𝐵𝜑) → 𝑥𝐴)))
1310, 12syl9 71 . . . . . . 7 (𝐴𝐵 → (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥(𝑥𝐴𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1413com23 77 . . . . . 6 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → (∃!𝑥(𝑥𝐵𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1514imp32 253 . . . . 5 ((𝐴𝐵 ∧ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑))) → ((𝑥𝐵𝜑) → 𝑥𝐴))
165, 15sylan2b 281 . . . 4 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1716expcomd 1375 . . 3 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → (𝜑 → (𝑥𝐵𝑥𝐴)))
1817imp 122 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐵𝑥𝐴))
192, 18impbid 127 1 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wex 1426  wcel 1438  ∃!weu 1948  wrex 2360  ∃!wreu 2361  wss 2997
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  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-rex 2365  df-reu 2366  df-in 3003  df-ss 3010
This theorem is referenced by:  supelti  6676
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