Theorem bdreu 15990

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 15992, and ⊢ (∀𝑥 ∈ V𝜑 ↔ ∀𝑥𝜑) (in minimal propositional calculus), so by bd0 15959, 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 15954	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3		ax-bdeq 15955	. . . . . 6 ⊢ BOUNDED 𝑥 = 𝑧
4	1, 3	ax-bdim 15949	. . . . 5 ⊢ BOUNDED (𝜑 → 𝑥 = 𝑧)
5	4	ax-bdal 15953	. . . 4 ⊢ BOUNDED ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
6	5	ax-bdex 15954	. . 3 ⊢ BOUNDED ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)
7	2, 6	ax-bdan 15950	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧))
8		reu3 2970	. 2 ⊢ (∃!𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 ∧ ∃𝑧 ∈ 𝑦 ∀𝑥 ∈ 𝑦 (𝜑 → 𝑥 = 𝑧)))
9	7, 8	bd0r 15960	1 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4   ∧ wa 104  ∀wral 2486  ∃wrex 2487  ∃!wreu 2488  BOUNDED wbd 15947

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-bd0 15948  ax-bdim 15949  ax-bdan 15950  ax-bdal 15953  ax-bdex 15954  ax-bdeq 15955

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-cleq 2200  df-clel 2203  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494

This theorem is referenced by:  bdrmo  15991