Theorem bdreu 13053

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 13055, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 13022, 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 13017	. . 3 |- BOUNDED E. x e. y ph
3		ax-bdeq 13018	. . . . . 6 |- BOUNDED x = z
4	1, 3	ax-bdim 13012	. . . . 5 |- BOUNDED ( ph -> x = z )
5	4	ax-bdal 13016	. . . 4 |- BOUNDED A. x e. y ( ph -> x = z )
6	5	ax-bdex 13017	. . 3 |- BOUNDED E. z e. y A. x e. y ( ph -> x = z )
7	2, 6	ax-bdan 13013	. 2 |- BOUNDED ( E. x e. y ph /\ E. z e. y A. x e. y ( ph -> x = z ) )
8		reu3 2874	. 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 13023	1 |- BOUNDED E! x e. y ph

Syntax hints:   -> wi 4   /\ wa 103   A.wral 2416   E.wrex 2417   E!wreu 2418  BOUNDED wbd 13010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-bd0 13011  ax-bdim 13012  ax-bdan 13013  ax-bdal 13016  ax-bdex 13017  ax-bdeq 13018

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-cleq 2132  df-clel 2135  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424

This theorem is referenced by:  bdrmo  13054