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Theorem reupick 3360
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 3091 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21ad2antrr 479 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
3 df-rex 2422 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2423 . . . . . 6 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
53, 4anbi12i 455 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑)))
61ancrd 324 . . . . . . . . . . 11 (𝐴𝐵 → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
76anim1d 334 . . . . . . . . . 10 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
8 an32 551 . . . . . . . . . 10 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∧ 𝑥𝐴))
97, 8syl6ib 160 . . . . . . . . 9 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
109eximdv 1852 . . . . . . . 8 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
11 eupick 2078 . . . . . . . . 9 ((∃!𝑥(𝑥𝐵𝜑) ∧ ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1211ex 114 . . . . . . . 8 (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴) → ((𝑥𝐵𝜑) → 𝑥𝐴)))
1310, 12syl9 72 . . . . . . 7 (𝐴𝐵 → (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥(𝑥𝐴𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1413com23 78 . . . . . 6 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → (∃!𝑥(𝑥𝐵𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1514imp32 255 . . . . 5 ((𝐴𝐵 ∧ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑))) → ((𝑥𝐵𝜑) → 𝑥𝐴))
165, 15sylan2b 285 . . . 4 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1716expcomd 1417 . . 3 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → (𝜑 → (𝑥𝐵𝑥𝐴)))
1817imp 123 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐵𝑥𝐴))
192, 18impbid 128 1 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wex 1468  wcel 1480  ∃!weu 1999  wrex 2417  ∃!wreu 2418  wss 3071
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-rex 2422  df-reu 2423  df-in 3077  df-ss 3084
This theorem is referenced by:  supelti  6889
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