Theorem bdreu 15347

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 15349, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 15316, 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 15311	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3		ax-bdeq 15312	. . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧
4	1, 3	ax-bdim 15306	. . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧)
5	4	ax-bdal 15310	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
6	5	ax-bdex 15311	. . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
7	2, 6	ax-bdan 15307	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))
8		reu3 2950	. 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)))
9	7, 8	bd0r 15317	1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4   ∧ wa 104  ∀wral 2472  ∃wrex 2473  ∃!wreu 2474  BOUNDED wbd 15304

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175  ax-bd0 15305  ax-bdim 15306  ax-bdan 15307  ax-bdal 15310  ax-bdex 15311  ax-bdeq 15312

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-cleq 2186  df-clel 2189  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480

This theorem is referenced by:  bdrmo  15348