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Theorem mosubop 4333
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 1319 . 2 yz∃*xφ
3 mosubopt 4332 . 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 1226   = wceq 1228  wex 1362  ∃*wmo 1883  cop 3353
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359
This theorem is referenced by:  ovi3  5560  ov6g  5561  oprabex3  5679
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