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Theorem eueq3 3011
Description: Equality has existential uniqueness (split into 3 cases). (Contributed by NM, 5-Apr-1995.) (Proof shortened by Mario Carneiro, 28-Sep-2015.)
Hypotheses
Ref Expression
eueq3.1
eueq3.2
eueq3.3
eueq3.4
Assertion
Ref Expression
eueq3
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem eueq3
StepHypRef Expression
1 eueq3.1 . . . 4
21eueq1 3009 . . 3
3 ibar 490 . . . . . 6
4 pm2.45 386 . . . . . . . . . 10
5 eueq3.4 . . . . . . . . . . . 12
65imnani 412 . . . . . . . . . . 11
76con2i 112 . . . . . . . . . 10
84, 7jaoi 368 . . . . . . . . 9
98con2i 112 . . . . . . . 8
104con2i 112 . . . . . . . . . 10
1110bianfd 892 . . . . . . . . 9
126bianfd 892 . . . . . . . . 9
1311, 12orbi12d 690 . . . . . . . 8
149, 13mtbid 291 . . . . . . 7
15 biorf 394 . . . . . . 7
1614, 15syl 15 . . . . . 6
173, 16bitrd 244 . . . . 5
18 3orrot 940 . . . . . 6
19 df-3or 935 . . . . . 6
2018, 19bitri 240 . . . . 5
2117, 20syl6bbr 254 . . . 4
2221eubidv 2212 . . 3
232, 22mpbii 202 . 2
24 eueq3.3 . . . 4
2524eueq1 3009 . . 3
26 ibar 490 . . . . . 6
276adantr 451 . . . . . . . . 9
28 pm2.46 387 . . . . . . . . . 10
2928adantr 451 . . . . . . . . 9
3027, 29jaoi 368 . . . . . . . 8
3130con2i 112 . . . . . . 7
32 biorf 394 . . . . . . 7
3331, 32syl 15 . . . . . 6
3426, 33bitrd 244 . . . . 5
35 df-3or 935 . . . . 5
3634, 35syl6bbr 254 . . . 4
3736eubidv 2212 . . 3
3825, 37mpbii 202 . 2
39 eueq3.2 . . . 4
4039eueq1 3009 . . 3
41 ibar 490 . . . . . 6
42 simpl 443 . . . . . . . . 9
43 simpl 443 . . . . . . . . 9
4442, 43orim12i 502 . . . . . . . 8
4544con3i 127 . . . . . . 7
46 biorf 394 . . . . . . 7
4745, 46syl 15 . . . . . 6
4841, 47bitrd 244 . . . . 5
49 3orcomb 944 . . . . . 6
50 df-3or 935 . . . . . 6
5149, 50bitri 240 . . . . 5
5248, 51syl6bbr 254 . . . 4
5352eubidv 2212 . . 3
5440, 53mpbii 202 . 2
5523, 38, 54ecase3 907 1
Colors of variables: wff setvar class
Syntax hints:   wn 3   wb 176   wo 357   wa 358   w3o 933   wceq 1642   wcel 1710  weu 2204  cvv 2859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861
This theorem is referenced by:  moeq3  3013
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