Theorem bdrmo 14796

Description: Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdrmo.1	⊢ BOUNDED 𝜑

Assertion

Ref	Expression
bdrmo	⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bdrmo

Step	Hyp	Ref	Expression
1		bdrmo.1	. . . 4 ⊢ BOUNDED 𝜑
2	1	ax-bdex 14759	. . 3 ⊢ BOUNDED ∃𝑥 ∈ 𝑦 𝜑
3	1	bdreu 14795	. . 3 ⊢ BOUNDED ∃!𝑥 ∈ 𝑦 𝜑
4	2, 3	ax-bdim 14754	. 2 ⊢ BOUNDED (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑)
5		rmo5 2693	. 2 ⊢ (∃*𝑥 ∈ 𝑦 𝜑 ↔ (∃𝑥 ∈ 𝑦 𝜑 → ∃!𝑥 ∈ 𝑦 𝜑))
6	4, 5	bd0r 14765	1 ⊢ BOUNDED ∃*𝑥 ∈ 𝑦 𝜑

Syntax hints:   → wi 4  ∃wrex 2456  ∃!wreu 2457  ∃*wrmo 2458  BOUNDED wbd 14752

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-bd0 14753  ax-bdim 14754  ax-bdan 14755  ax-bdal 14758  ax-bdex 14759  ax-bdeq 14760

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-cleq 2170  df-clel 2173  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463

This theorem is referenced by: (None)