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Metamath Proof Explorer

Theorem eueq2

Description: Equality has existential uniqueness (split into 2 cases). (Contributed by NM, 5-Apr-1995)

Ref Expression
Hypotheses eueq2.1 
|- A e. _V
eueq2.2 
|- B e. _V
Assertion eueq2 
|- E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) )

Proof

Step Hyp Ref Expression
1 eueq2.1 
 |-  A e. _V
2 eueq2.2 
 |-  B e. _V
3 notnot 
 |-  ( ph -> -. -. ph )
4 1 eueqi 
 |-  E! x x = A
5 euanv 
 |-  ( E! x ( ph /\ x = A ) <-> ( ph /\ E! x x = A ) )
6 5 biimpri 
 |-  ( ( ph /\ E! x x = A ) -> E! x ( ph /\ x = A ) )
7 4 6 mpan2 
 |-  ( ph -> E! x ( ph /\ x = A ) )
8 euorv 
 |-  ( ( -. -. ph /\ E! x ( ph /\ x = A ) ) -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
9 3 7 8 syl2anc 
 |-  ( ph -> E! x ( -. ph \/ ( ph /\ x = A ) ) )
10 orcom 
 |-  ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ -. ph ) )
11 3 bianfd 
 |-  ( ph -> ( -. ph <-> ( -. ph /\ x = B ) ) )
12 11 orbi2d 
 |-  ( ph -> ( ( ( ph /\ x = A ) \/ -. ph ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
13 10 12 bitrid 
 |-  ( ph -> ( ( -. ph \/ ( ph /\ x = A ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
14 13 eubidv 
 |-  ( ph -> ( E! x ( -. ph \/ ( ph /\ x = A ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
15 9 14 mpbid 
 |-  ( ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
16 2 eueqi 
 |-  E! x x = B
17 euanv 
 |-  ( E! x ( -. ph /\ x = B ) <-> ( -. ph /\ E! x x = B ) )
18 17 biimpri 
 |-  ( ( -. ph /\ E! x x = B ) -> E! x ( -. ph /\ x = B ) )
19 16 18 mpan2 
 |-  ( -. ph -> E! x ( -. ph /\ x = B ) )
20 euorv 
 |-  ( ( -. ph /\ E! x ( -. ph /\ x = B ) ) -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
21 19 20 mpdan 
 |-  ( -. ph -> E! x ( ph \/ ( -. ph /\ x = B ) ) )
22 id 
 |-  ( -. ph -> -. ph )
23 22 bianfd 
 |-  ( -. ph -> ( ph <-> ( ph /\ x = A ) ) )
24 23 orbi1d 
 |-  ( -. ph -> ( ( ph \/ ( -. ph /\ x = B ) ) <-> ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
25 24 eubidv 
 |-  ( -. ph -> ( E! x ( ph \/ ( -. ph /\ x = B ) ) <-> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) ) )
26 21 25 mpbid 
 |-  ( -. ph -> E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) ) )
27 15 26 pm2.61i 
 |-  E! x ( ( ph /\ x = A ) \/ ( -. ph /\ x = B ) )