Theorem bdrmo 11747

Description: Boundedness of existential at-most-one. (Contributed by BJ, 16-Oct-2019.)

Hypothesis

Ref	Expression
bdrmo.1	|- BOUNDED ph

Assertion

Ref	Expression
bdrmo	|- BOUNDED E* x e. y ph

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)

Proof of Theorem bdrmo

Step	Hyp	Ref	Expression
1		bdrmo.1	. . . 4 |- BOUNDED ph
2	1	ax-bdex 11710	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 11746	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 11705	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2582	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 11716	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2360   E!wreu 2361   E*wrmo 2362  BOUNDED wbd 11703

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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-bd0 11704  ax-bdim 11705  ax-bdan 11706  ax-bdal 11709  ax-bdex 11710  ax-bdeq 11711

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-cleq 2081  df-clel 2084  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367

This theorem is referenced by: (None)