Theorem mosubop 4798

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 1499	. 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑
3		mosubopt 4797	. 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 104  ∀wal 1396   = wceq 1398  ∃wex 1541  ∃*wmo 2080  ⟨cop 3676

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682

This theorem is referenced by:  ovi3  6169  ov6g  6170  oprabex3  6300  axaddf  8148  axmulf  8149