Theorem bdreu 16881

Description: Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula ∀𝑥 ∈ 𝐴𝜑 need not be bounded even if 𝐴 and 𝜑 are. Indeed, V is bounded by bdcvv 16883, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 16850, if ∀𝑥 ∈ V𝜑 were bounded when 𝜑 is bounded, then ∀𝑥𝜑 would be bounded as well when 𝜑 is bounded, which is not the case. The same remark holds with ∃, ∃!, ∃*. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdreu.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdreu	⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem bdreu

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdreu.1	. . . 4 ⊢ BOUNDED 𝜑
2	1	ax-bdex 16845	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3		ax-bdeq 16846	. . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧
4	1, 3	ax-bdim 16840	. . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧)
5	4	ax-bdal 16844	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
6	5	ax-bdex 16845	. . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
7	2, 6	ax-bdan 16841	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))
8		reu3 3016	. 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)))
9	7, 8	bd0r 16851	1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑

This proof depends on syntax axioms:   → wi 4   ∧ wa 104  ∀wral 2528  ∃wrex 2529  ∃!wreu 2530  BOUNDED wbd 16838

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16839  ax-bdim 16840  ax-bdan 16841  ax-bdal 16844  ax-bdex 16845  ax-bdeq 16846

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-cleq 2231  df-clel 2234  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536

This theorem is used by:  bdrmo  16882