Theorem rmoi 2997

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmoi.b	|- ( x = B -> ( ph <-> ps ) )
rmoi.c	|- ( x = C -> ( ph <-> ch ) )

Assertion

Ref	Expression
rmoi	|- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )

Distinct variable groups:   x, A   x, B   x, C   ps, x   ch, x

Allowed substitution hint:   ph( x)

Proof of Theorem rmoi

Step	Hyp	Ref	Expression
1		rmoi.b	. . 3 |- ( x = B -> ( ph <-> ps ) )
2		rmoi.c	. . 3 |- ( x = C -> ( ph <-> ch ) )
3	1, 2	rmob 2996	. 2 |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
4	3	biimp3ar 1324	1 |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   E*wrmo 2417

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 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-rmo 2422  df-v 2683

This theorem is referenced by: (None)