Theorem bdreu 16893

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 16895, and |- ( A. x e. _V ph <-> A. x ph ) (in minimal propositional calculus), so by bd0 16862, 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 16857	. . 3 |- BOUNDED E. x e. y ph
3		ax-bdeq 16858	. . . . . 6 |- BOUNDED x = z
4	1, 3	ax-bdim 16852	. . . . 5 |- BOUNDED ( ph -> x = z )
5	4	ax-bdal 16856	. . . 4 |- BOUNDED A. x e. y ( ph -> x = z )
6	5	ax-bdex 16857	. . 3 |- BOUNDED E. z e. y A. x e. y ( ph -> x = z )
7	2, 6	ax-bdan 16853	. 2 |- BOUNDED ( E. x e. y ph /\ E. z e. y A. x e. y ( ph -> x = z ) )
8		reu3 3016	. 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 16863	1 |- BOUNDED E! x e. y ph

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   A.wral 2528   E.wrex 2529   E!wreu 2530  BOUNDED wbd 16850

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-bd0 16851  ax-bdim 16852  ax-bdan 16853  ax-bdal 16856  ax-bdex 16857  ax-bdeq 16858

This proof depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-cleq 2231  df-clel 2234  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536

This theorem is used by:  bdrmo  16894