Theorem bdreu 13890

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 13892, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 13859, 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 13854	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3		ax-bdeq 13855	. . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧
4	1, 3	ax-bdim 13849	. . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧)
5	4	ax-bdal 13853	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
6	5	ax-bdex 13854	. . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
7	2, 6	ax-bdan 13850	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))
8		reu3 2920	. 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)))
9	7, 8	bd0r 13860	1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4   ∧ wa 103  ∀wral 2448  ∃wrex 2449  ∃!wreu 2450  BOUNDED wbd 13847

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-bd0 13848  ax-bdim 13849  ax-bdan 13850  ax-bdal 13853  ax-bdex 13854  ax-bdeq 13855

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163  df-clel 2166  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456

This theorem is referenced by:  bdrmo  13891