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Theorem reupick 3870
 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 3562 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21ad2antrr 758 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
3 df-rex 2902 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 2903 . . . . . 6 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
53, 4anbi12i 729 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑)))
61ancrd 575 . . . . . . . . . . 11 (𝐴𝐵 → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
76anim1d 586 . . . . . . . . . 10 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
8 an32 835 . . . . . . . . . 10 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∧ 𝑥𝐴))
97, 8syl6ib 240 . . . . . . . . 9 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
109eximdv 1833 . . . . . . . 8 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
11 eupick 2524 . . . . . . . . 9 ((∃!𝑥(𝑥𝐵𝜑) ∧ ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1211ex 449 . . . . . . . 8 (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴) → ((𝑥𝐵𝜑) → 𝑥𝐴)))
1310, 12syl9 75 . . . . . . 7 (𝐴𝐵 → (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥(𝑥𝐴𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1413com23 84 . . . . . 6 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → (∃!𝑥(𝑥𝐵𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1514imp32 448 . . . . 5 ((𝐴𝐵 ∧ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑))) → ((𝑥𝐵𝜑) → 𝑥𝐴))
165, 15sylan2b 491 . . . 4 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1716expcomd 453 . . 3 ((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) → (𝜑 → (𝑥𝐵𝑥𝐴)))
1817imp 444 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐵𝑥𝐴))
192, 18impbid 201 1 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  ∃wrex 2897  ∃!wreu 2898   ⊆ wss 3540 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-rex 2902  df-reu 2903  df-in 3547  df-ss 3554 This theorem is referenced by: (None)
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