Theorem bdreu 9975

Description: Boundedness of existential uniqueness.

Remark regarding restricted quantifiers: the formula A. x e. A ph need not be bounded even if A and ph are. Indeed, _V is bounded by bdcvv 9977, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 9944, if A. x e. _V ph were bounded when ph is bounded, then A. x ph would be bounded as well when ph is bounded, which is not the case. The same remark holds with E. , E! , E*. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdreu.1	|- BOUNDED ph

Assertion

Ref	Expression
bdreu	|- BOUNDED E! x e. y ph

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bdreu

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdreu.1	. . . 4 |- BOUNDED ph
2	1	ax-bdex 9939	. . 3 |- BOUNDED E. x e. y ph
3		ax-bdeq 9940	. . . . . 6 |- BOUNDED x = z
4	1, 3	ax-bdim 9934	. . . . 5 |- BOUNDED ( ph -> x = z )
5	4	ax-bdal 9938	. . . 4 |- BOUNDED A. x e. y ( ph -> x = z )
6	5	ax-bdex 9939	. . 3 |- BOUNDED E. z e. y A. x e. y ( ph -> x = z )
7	2, 6	ax-bdan 9935	. 2 |- BOUNDED ( E. x e. y ph /\ E. z e. y A. x e. y ( ph -> x = z ) )
8		reu3 2731	. 2 |- ( E! x e. y ph <-> ( E. x e. y ph /\ E. z e. y A. x e. y ( ph -> x = z ) ) )
9	7, 8	bd0r 9945	1 |- BOUNDED E! x e. y ph

Syntax hints:   -> wi 4   /\ wa 97   A.wral 2306   E.wrex 2307   E!wreu 2308  BOUNDED wbd 9932

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bd0 9933  ax-bdim 9934  ax-bdan 9935  ax-bdal 9938  ax-bdex 9939  ax-bdeq 9940

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-cleq 2033  df-clel 2036  df-ral 2311  df-rex 2312  df-reu 2313  df-rmo 2314

This theorem is referenced by:  bdrmo  9976