Theorem mosubop 5496

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 1826	. 2 ⊢ ∀𝑦∀𝑧∃*𝑥𝜑
3		mosubopt 5495	. 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 400  ∀wal 1568   = wceq 1570  ∃wex 1809  ∃*wmo 2565  ⟨cop 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597

This theorem is referenced by:  ov3  7575  ov6g  7576  oprabex3  7975  axaddf  11131  axmulf  11132