Theorem bdreu 16176

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 16178, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 16145, 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 16140	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3		ax-bdeq 16141	. . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧
4	1, 3	ax-bdim 16135	. . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧)
5	4	ax-bdal 16139	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
6	5	ax-bdex 16140	. . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
7	2, 6	ax-bdan 16136	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))
8		reu3 2993	. 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)))
9	7, 8	bd0r 16146	1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4   ∧ wa 104  ∀wral 2508  ∃wrex 2509  ∃!wreu 2510  BOUNDED wbd 16133

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-bd0 16134  ax-bdim 16135  ax-bdan 16136  ax-bdal 16139  ax-bdex 16140  ax-bdeq 16141

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-cleq 2222  df-clel 2225  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516

This theorem is referenced by:  bdrmo  16177