Theorem rmoi 3068

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 3067	. 2 |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) ) -> ( B = C <-> ( C e. A /\ ch ) ) )
4	3	biimp3ar 1356	1 |- ( ( E* x e. A ph /\ ( B e. A /\ ps ) /\ ( C e. A /\ ch ) ) -> B = C )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 979   = wceq 1363   e. wcel 2158   E*wrmo 2468

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 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-rmo 2473  df-v 2751

This theorem is referenced by: (None)