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Theorem mosubop 4694
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 1450 . 2 𝑦𝑧∃*𝑥𝜑
3 mosubopt 4693 . 2 (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
42, 3ax-mp 5 1 ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
Colors of variables: wff set class
Syntax hints:  wa 104  wal 1351   = wceq 1353  wex 1492  ∃*wmo 2027  cop 3597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603
This theorem is referenced by:  ovi3  6013  ov6g  6014  oprabex3  6132  axaddf  7869  axmulf  7870
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