Theorem bdreu 10646

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 10648, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 10615, 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 10610	. . 3 |- BOUNDED E. x e. y ph
3		ax-bdeq 10611	. . . . . 6 |- BOUNDED x = z
4	1, 3	ax-bdim 10605	. . . . 5 |- BOUNDED ( ph -> x = z )
5	4	ax-bdal 10609	. . . 4 |- BOUNDED A. x e. y ( ph -> x = z )
6	5	ax-bdex 10610	. . 3 |- BOUNDED E. z e. y A. x e. y ( ph -> x = z )
7	2, 6	ax-bdan 10606	. 2 |- BOUNDED ( E. x e. y ph /\ E. z e. y A. x e. y ( ph -> x = z ) )
8		reu3 2782	. 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 10616	1 |- BOUNDED E! x e. y ph

Syntax hints:   -> wi 4   /\ wa 102   A.wral 2348   E.wrex 2349   E!wreu 2350  BOUNDED wbd 10603

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-bd0 10604  ax-bdim 10605  ax-bdan 10606  ax-bdal 10609  ax-bdex 10610  ax-bdeq 10611

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-cleq 2074  df-clel 2077  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356

This theorem is referenced by:  bdrmo  10647