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Theorem moeq3 3111
 Description: "At most one" property of equality (split into 3 cases). (The first 2 hypotheses could be eliminated with longer proof.) (Contributed by NM, 23-Apr-1995.)
Hypotheses
Ref Expression
moeq3.1
moeq3.2
moeq3.3
Assertion
Ref Expression
moeq3
Distinct variable groups:   ,   ,   ,   ,   ,

Proof of Theorem moeq3
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2445 . . . . . . 7
21anbi2d 685 . . . . . 6
3 biidd 229 . . . . . 6
4 biidd 229 . . . . . 6
52, 3, 43orbi123d 1253 . . . . 5
65eubidv 2289 . . . 4
7 vex 2959 . . . . 5
8 moeq3.1 . . . . 5
9 moeq3.2 . . . . 5
10 moeq3.3 . . . . 5
117, 8, 9, 10eueq3 3109 . . . 4
126, 11vtoclg 3011 . . 3
13 eumo 2321 . . 3
1412, 13syl 16 . 2
15 vex 2959 . . . . . . . . 9
16 eleq1 2496 . . . . . . . . 9
1715, 16mpbii 203 . . . . . . . 8
18 pm2.21 102 . . . . . . . 8
1917, 18syl5 30 . . . . . . 7
2019anim2d 549 . . . . . 6
2120orim1d 813 . . . . 5
22 3orass 939 . . . . 5
23 3orass 939 . . . . 5
2421, 22, 233imtr4g 262 . . . 4
2524alrimiv 1641 . . 3
26 euimmo 2330 . . 3
2725, 11, 26ee10 1385 . 2
2814, 27pm2.61i 158 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wo 358   wa 359   w3o 935  wal 1549   wceq 1652   wcel 1725  weu 2281  wmo 2282  cvv 2956 This theorem is referenced by:  tz7.44lem1  6663 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958
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