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Theorem reupick 3215
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by NM, 21-Aug-1999.)
Assertion
Ref Expression
reupick (((AB (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 2933 . . 3 (AB → (x Ax B))
21ad2antrr 457 . 2 (((AB (x A φ ∃!x B φ)) φ) → (x Ax B))
3 df-rex 2306 . . . . . 6 (x A φx(x A φ))
4 df-reu 2307 . . . . . 6 (∃!x B φ∃!x(x B φ))
53, 4anbi12i 433 . . . . 5 ((x A φ ∃!x B φ) ↔ (x(x A φ) ∃!x(x B φ)))
61ancrd 309 . . . . . . . . . . 11 (AB → (x A → (x B x A)))
76anim1d 319 . . . . . . . . . 10 (AB → ((x A φ) → ((x B x A) φ)))
8 an32 496 . . . . . . . . . 10 (((x B x A) φ) ↔ ((x B φ) x A))
97, 8syl6ib 150 . . . . . . . . 9 (AB → ((x A φ) → ((x B φ) x A)))
109eximdv 1757 . . . . . . . 8 (AB → (x(x A φ) → x((x B φ) x A)))
11 eupick 1976 . . . . . . . . 9 ((∃!x(x B φ) x((x B φ) x A)) → ((x B φ) → x A))
1211ex 108 . . . . . . . 8 (∃!x(x B φ) → (x((x B φ) x A) → ((x B φ) → x A)))
1310, 12syl9 66 . . . . . . 7 (AB → (∃!x(x B φ) → (x(x A φ) → ((x B φ) → x A))))
1413com23 72 . . . . . 6 (AB → (x(x A φ) → (∃!x(x B φ) → ((x B φ) → x A))))
1514imp32 244 . . . . 5 ((AB (x(x A φ) ∃!x(x B φ))) → ((x B φ) → x A))
165, 15sylan2b 271 . . . 4 ((AB (x A φ ∃!x B φ)) → ((x B φ) → x A))
1716expcomd 1327 . . 3 ((AB (x A φ ∃!x B φ)) → (φ → (x Bx A)))
1817imp 115 . 2 (((AB (x A φ ∃!x B φ)) φ) → (x Bx A))
192, 18impbid 120 1 (((AB (x A φ ∃!x B φ)) φ) → (x Ax B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wex 1378   wcel 1390  ∃!weu 1897  wrex 2301  ∃!wreu 2302  wss 2911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-rex 2306  df-reu 2307  df-in 2918  df-ss 2925
This theorem is referenced by: (None)
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