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Theorem reupick2 3217
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 15-Dec-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick2 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reupick2
StepHypRef Expression
1 ancr 304 . . . . . 6 ((ψφ) → (ψ → (φ ψ)))
21ralimi 2378 . . . . 5 (x A (ψφ) → x A (ψ → (φ ψ)))
3 rexim 2407 . . . . 5 (x A (ψ → (φ ψ)) → (x A ψx A (φ ψ)))
42, 3syl 14 . . . 4 (x A (ψφ) → (x A ψx A (φ ψ)))
5 reupick3 3216 . . . . . 6 ((∃!x A φ x A (φ ψ) x A) → (φψ))
653exp 1102 . . . . 5 (∃!x A φ → (x A (φ ψ) → (x A → (φψ))))
76com12 27 . . . 4 (x A (φ ψ) → (∃!x A φ → (x A → (φψ))))
84, 7syl6 29 . . 3 (x A (ψφ) → (x A ψ → (∃!x A φ → (x A → (φψ)))))
983imp1 1116 . 2 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
10 rsp 2363 . . . 4 (x A (ψφ) → (x A → (ψφ)))
11103ad2ant1 924 . . 3 ((x A (ψφ) x A ψ ∃!x A φ) → (x A → (ψφ)))
1211imp 115 . 2 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (ψφ))
139, 12impbid 120 1 (((x A (ψφ) x A ψ ∃!x A φ) x A) → (φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   w3a 884   wcel 1390  wral 2300  wrex 2301  ∃!wreu 2302
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-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425
This theorem depends on definitions:  df-bi 110  df-3an 886  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-ral 2305  df-rex 2306  df-reu 2307
This theorem is referenced by: (None)
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