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Theorem reupick 3248
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 2966 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21ad2antrr 465 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
3 df-rex 2329 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2330 . . . . . 6 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
53, 4anbi12i 441 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑)))
61ancrd 313 . . . . . . . . . . 11 (𝐴𝐵 → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
76anim1d 323 . . . . . . . . . 10 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
8 an32 504 . . . . . . . . . 10 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∧ 𝑥𝐴))
97, 8syl6ib 154 . . . . . . . . 9 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
109eximdv 1776 . . . . . . . 8 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
11 eupick 1995 . . . . . . . . 9 ((∃!𝑥(𝑥𝐵𝜑) ∧ ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1211ex 112 . . . . . . . 8 (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴) → ((𝑥𝐵𝜑) → 𝑥𝐴)))
1310, 12syl9 70 . . . . . . 7 (𝐴𝐵 → (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥(𝑥𝐴𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1413com23 76 . . . . . 6 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → (∃!𝑥(𝑥𝐵𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1514imp32 248 . . . . 5 ((𝐴𝐵 ∧ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑))) → ((𝑥𝐵𝜑) → 𝑥𝐴))
165, 15sylan2b 275 . . . 4 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1716expcomd 1346 . . 3 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → (𝜑 → (𝑥𝐵𝑥𝐴)))
1817imp 119 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐵𝑥𝐴))
192, 18impbid 124 1 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wex 1397  wcel 1409  ∃!weu 1916  wrex 2324  ∃!wreu 2325  wss 2944
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-rex 2329  df-reu 2330  df-in 2951  df-ss 2958
This theorem is referenced by: (None)
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