Theorem bdreu 16450

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 16452, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 16419, 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 16414	. . 3 |- BOUNDED E. x e. y ph
3		ax-bdeq 16415	. . . . . 6 |- BOUNDED x = z
4	1, 3	ax-bdim 16409	. . . . 5 |- BOUNDED ( ph -> x = z )
5	4	ax-bdal 16413	. . . 4 |- BOUNDED A. x e. y ( ph -> x = z )
6	5	ax-bdex 16414	. . 3 |- BOUNDED E. z e. y A. x e. y ( ph -> x = z )
7	2, 6	ax-bdan 16410	. 2 |- BOUNDED ( E. x e. y ph /\ E. z e. y A. x e. y ( ph -> x = z ) )
8		reu3 2996	. 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 16420	1 |- BOUNDED E! x e. y ph

Syntax hints:   -> wi 4   /\ wa 104   A.wral 2510   E.wrex 2511   E!wreu 2512  BOUNDED wbd 16407

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213  ax-bd0 16408  ax-bdim 16409  ax-bdan 16410  ax-bdal 16413  ax-bdex 16414  ax-bdeq 16415

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-cleq 2224  df-clel 2227  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518

This theorem is referenced by:  bdrmo  16451