Theorem mosubop 5366

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 1798	. 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑
3		mosubopt 5365	. 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 399  ∀wal 1536   = wceq 1538  ∃wex 1781  ∃*wmo 2596  ⟨cop 4531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532

This theorem is referenced by:  ov3  7291  ov6g  7292  oprabex3  7660  axaddf  10556  axmulf  10557