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Theorem reupick 4054
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 3738 . . 3 (𝐴𝐵 → (𝑥𝐴𝑥𝐵))
21ad2antrr 764 . 2 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
3 df-rex 3056 . . . . . 6 (∃𝑥𝐴 𝜑 ↔ ∃𝑥(𝑥𝐴𝜑))
4 df-reu 3057 . . . . . 6 (∃!𝑥𝐵 𝜑 ↔ ∃!𝑥(𝑥𝐵𝜑))
53, 4anbi12i 735 . . . . 5 ((∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑) ↔ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑)))
61ancrd 578 . . . . . . . . . . 11 (𝐴𝐵 → (𝑥𝐴 → (𝑥𝐵𝑥𝐴)))
76anim1d 589 . . . . . . . . . 10 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝑥𝐴) ∧ 𝜑)))
8 an32 874 . . . . . . . . . 10 (((𝑥𝐵𝑥𝐴) ∧ 𝜑) ↔ ((𝑥𝐵𝜑) ∧ 𝑥𝐴))
97, 8syl6ib 241 . . . . . . . . 9 (𝐴𝐵 → ((𝑥𝐴𝜑) → ((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
109eximdv 1995 . . . . . . . 8 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)))
11 eupick 2674 . . . . . . . . 9 ((∃!𝑥(𝑥𝐵𝜑) ∧ ∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴)) → ((𝑥𝐵𝜑) → 𝑥𝐴))
1211ex 449 . . . . . . . 8 (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥((𝑥𝐵𝜑) ∧ 𝑥𝐴) → ((𝑥𝐵𝜑) → 𝑥𝐴)))
1310, 12syl9 77 . . . . . . 7 (𝐴𝐵 → (∃!𝑥(𝑥𝐵𝜑) → (∃𝑥(𝑥𝐴𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1413com23 86 . . . . . 6 (𝐴𝐵 → (∃𝑥(𝑥𝐴𝜑) → (∃!𝑥(𝑥𝐵𝜑) → ((𝑥𝐵𝜑) → 𝑥𝐴))))
1514imp32 448 . . . . 5 ((𝐴𝐵 ∧ (∃𝑥(𝑥𝐴𝜑) ∧ ∃!𝑥(𝑥𝐵𝜑))) → ((𝑥𝐵𝜑) → 𝑥𝐴))
165, 15sylan2b 493 . . . 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 202 1 (((𝐴𝐵 ∧ (∃𝑥𝐴 𝜑 ∧ ∃!𝑥𝐵 𝜑)) ∧ 𝜑) → (𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wex 1853  wcel 2139  ∃!weu 2607  wrex 3051  ∃!wreu 3052  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-rex 3056  df-reu 3057  df-in 3722  df-ss 3729
This theorem is referenced by: (None)
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