Theorem mosubop 5425

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	⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem mosubop

Step	Hyp	Ref	Expression
1		mosubop.1	. . 3 ⊢ ∃*𝑥𝜑
2	1	gen2 1799	. 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑
3		mosubopt 5424	. 2 ⊢ (∀𝑦∀𝑧∃*𝑥𝜑 → ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑))
4	2, 3	ax-mp 5	1 ⊢ ∃*𝑥∃𝑦∃𝑧(𝐴 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)

Syntax hints:   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782  ∃*wmo 2538  ⟨cop 4567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568

This theorem is referenced by:  ov3  7435  ov6g  7436  oprabex3  7820  axaddf  10901  axmulf  10902