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Theorem reupick 3539
 Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reupick (((A B (x A φ ∃!x B φ)) φ) → (x Ax B))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   φ(x)

Proof of Theorem reupick
StepHypRef Expression
1 ssel 3267 . . 3 (A B → (x Ax B))
21ad2antrr 706 . 2 (((A B (x A φ ∃!x B φ)) φ) → (x Ax B))
3 df-rex 2620 . . . . . 6 (x A φx(x A φ))
4 df-reu 2621 . . . . . 6 (∃!x B φ∃!x(x B φ))
53, 4anbi12i 678 . . . . 5 ((x A φ ∃!x B φ) ↔ (x(x A φ) ∃!x(x B φ)))
61ancrd 537 . . . . . . . . . . 11 (A B → (x A → (x B x A)))
76anim1d 547 . . . . . . . . . 10 (A B → ((x A φ) → ((x B x A) φ)))
8 an32 773 . . . . . . . . . 10 (((x B x A) φ) ↔ ((x B φ) x A))
97, 8syl6ib 217 . . . . . . . . 9 (A B → ((x A φ) → ((x B φ) x A)))
109eximdv 1622 . . . . . . . 8 (A B → (x(x A φ) → x((x B φ) x A)))
11 eupick 2267 . . . . . . . . 9 ((∃!x(x B φ) x((x B φ) x A)) → ((x B φ) → x A))
1211ex 423 . . . . . . . 8 (∃!x(x B φ) → (x((x B φ) x A) → ((x B φ) → x A)))
1310, 12syl9 66 . . . . . . 7 (A B → (∃!x(x B φ) → (x(x A φ) → ((x B φ) → x A))))
1413com23 72 . . . . . 6 (A B → (x(x A φ) → (∃!x(x B φ) → ((x B φ) → x A))))
1514imp32 422 . . . . 5 ((A B (x(x A φ) ∃!x(x B φ))) → ((x B φ) → x A))
165, 15sylan2b 461 . . . 4 ((A B (x A φ ∃!x B φ)) → ((x B φ) → x A))
1716exp3acom23 1372 . . 3 ((A B (x A φ ∃!x B φ)) → (φ → (x Bx A)))
1817imp 418 . 2 (((A B (x A φ ∃!x B φ)) φ) → (x Bx A))
192, 18impbid 183 1 (((A B (x A φ ∃!x B φ)) φ) → (x Ax B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  ∃!weu 2204  ∃wrex 2615  ∃!wreu 2616   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-reu 2621  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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