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Theorem mosubop 5521
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 1793 . 2 𝑦𝑧∃*𝑥𝜑
3 mosubopt 5520 . 2 (∀𝑦𝑧∃*𝑥𝜑 → ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
42, 3ax-mp 5 1 ∃*𝑥𝑦𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wa 395  wal 1535   = wceq 1537  wex 1776  ∃*wmo 2536  cop 4637
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638
This theorem is referenced by:  ov3  7596  ov6g  7597  oprabex3  8001  axaddf  11183  axmulf  11184
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