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Theorem reupick3 3216
Description: Restricted uniqueness "picks" a member of a subclass. (Contributed by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reupick3 ((∃!x A φ x A (φ ψ) x A) → (φψ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem reupick3
StepHypRef Expression
1 df-reu 2307 . . . 4 (∃!x A φ∃!x(x A φ))
2 df-rex 2306 . . . . 5 (x A (φ ψ) ↔ x(x A (φ ψ)))
3 anass 381 . . . . . 6 (((x A φ) ψ) ↔ (x A (φ ψ)))
43exbii 1493 . . . . 5 (x((x A φ) ψ) ↔ x(x A (φ ψ)))
52, 4bitr4i 176 . . . 4 (x A (φ ψ) ↔ x((x A φ) ψ))
6 eupick 1976 . . . 4 ((∃!x(x A φ) x((x A φ) ψ)) → ((x A φ) → ψ))
71, 5, 6syl2anb 275 . . 3 ((∃!x A φ x A (φ ψ)) → ((x A φ) → ψ))
87expd 245 . 2 ((∃!x A φ x A (φ ψ)) → (x A → (φψ)))
983impia 1100 1 ((∃!x A φ x A (φ ψ) x A) → (φψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   w3a 884  wex 1378   wcel 1390  ∃!weu 1897  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-rex 2306  df-reu 2307
This theorem is referenced by:  reupick2  3217
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