Theorem bdrmo 15792

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 15755	. . 3 |- BOUNDED E. x e. y ph
3	1	bdreu 15791	. . 3 |- BOUNDED E! x e. y ph
4	2, 3	ax-bdim 15750	. 2 |- BOUNDED ( E. x e. y ph -> E! x e. y ph )
5		rmo5 2726	. 2 |- ( E* x e. y ph <-> ( E. x e. y ph -> E! x e. y ph ) )
6	4, 5	bd0r 15761	1 |- BOUNDED E* x e. y ph

Syntax hints:   -> wi 4   E.wrex 2485   E!wreu 2486   E*wrmo 2487  BOUNDED wbd 15748

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-bd0 15749  ax-bdim 15750  ax-bdan 15751  ax-bdal 15754  ax-bdex 15755  ax-bdeq 15756

This theorem depends on definitions:  df-bi 117  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-cleq 2198  df-clel 2201  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492

This theorem is referenced by: (None)