Theorem bdrmo 16796

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 16759	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 16795	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 16754	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2773	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 16765	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2529   E!wreu 2530   E*wrmo 2531  BOUNDED wbd 16752

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 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 16753  ax-bdim 16754  ax-bdan 16755  ax-bdal 16758  ax-bdex 16759  ax-bdeq 16760

This theorem 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 referenced by: (None)