Theorem moeq3dc 2925

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 2923	. 2 |- (DECID ph -> (DECID ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
6		eumo 2068	. 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 104   \/ wo 709  DECID wdc 835   \/ w3o 978   = wceq 1363   E!weu 2036   E*wmo 2037   e. wcel 2158   _Vcvv 2749

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-in1 615  ax-in2 616  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-dc 836  df-3or 980  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-v 2751

This theorem is referenced by: (None)