Theorem moeq3dc 2906

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 2904	. 2 |- (DECID ph -> (DECID ps -> E! x ( ( ph /\ x = A ) \/ ( -. ( ph \/ ps ) /\ x = B ) \/ ( ps /\ x = C ) ) ) )
6		eumo 2051	. 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 103   \/ wo 703  DECID wdc 829   \/ w3o 972   = wceq 1348   E!weu 2019   E*wmo 2020   e. wcel 2141   _Vcvv 2730

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-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by: (None)