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Theorem mosubop 4349
Description: "At most one" remains true inside ordered pair quantification. (Contributed by NM, 28-May-1995.)
Hypothesis
Ref Expression
mosubop.1 ∃*xφ
Assertion
Ref Expression
mosubop ∃*xyz(A = ⟨y, z φ)
Distinct variable group:   x,y,z,A
Allowed substitution hints:   φ(x,y,z)

Proof of Theorem mosubop
StepHypRef Expression
1 mosubop.1 . . 3 ∃*xφ
21gen2 1336 . 2 yz∃*xφ
3 mosubopt 4348 . 2 (yz∃*xφ∃*xyz(A = ⟨y, z φ))
42, 3ax-mp 7 1 ∃*xyz(A = ⟨y, z φ)
Colors of variables: wff set class
Syntax hints:   wa 97  wal 1240   = wceq 1242  wex 1378  ∃*wmo 1898  cop 3370
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  ovi3  5579  ov6g  5580  oprabex3  5698
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