Theorem moeq3dc 2768

Description: "At most one" property of equality (split into 3 cases). (Contributed by Jim Kingdon, 7-Jul-2018.)

Hypotheses

Ref	Expression
moeq3dc.1	|- A e. _V
moeq3dc.2	|- B e. _V
moeq3dc.3	|- C e. _V
moeq3dc.4	|- -. ( ph /\ ps )

Assertion

Ref	Expression
moeq3dc	|- (DECID ph -> (DECID ps -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )

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

Proof of Theorem moeq3dc

Step	Hyp	Ref	Expression
1		moeq3dc.1	. . 3 |- A e. _V
2		moeq3dc.2	. . 3 |- B e. _V
3		moeq3dc.3	. . 3 |- C e. _V
4		moeq3dc.4	. . 3 |- -. ( ph /\ ps )
5	1, 2, 3, 4	eueq3dc 2766	. 2 |- (DECID ph -> (DECID ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
6		eumo 1973	. 2 |- ( E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) )
7	5, 6	syl6 33	1 |- (DECID ph -> (DECID ps -> E* x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   \/ wo 661  DECID wdc 775   \/ w3o 918   = wceq 1284   e. wcel 1433   E!weu 1941   E*wmo 1942   _Vcvv 2601

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-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-v 2603

This theorem is referenced by: (None)