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Theorem moeq3 3276
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 2472 . . . . . . 7
21anbi2d 703 . . . . . 6
3 biidd 237 . . . . . 6
4 biidd 237 . . . . . 6
52, 3, 43orbi123d 1298 . . . . 5
65eubidv 2304 . . . 4
7 vex 3112 . . . . 5
8 moeq3.1 . . . . 5
9 moeq3.2 . . . . 5
10 moeq3.3 . . . . 5
117, 8, 9, 10eueq3 3274 . . . 4
126, 11vtoclg 3167 . . 3
13 eumo 2313 . . 3
1412, 13syl 16 . 2
15 eqvisset 3117 . . . . . . . 8
16 pm2.21 108 . . . . . . . 8
1715, 16syl5 32 . . . . . . 7
1817anim2d 565 . . . . . 6
1918orim1d 839 . . . . 5
20 3orass 976 . . . . 5
21 3orass 976 . . . . 5
2219, 20, 213imtr4g 270 . . . 4
2322alrimiv 1719 . . 3
24 euimmo 2343 . . 3
2523, 11, 24mpisyl 18 . 2
2614, 25pm2.61i 164 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  \/w3o 972  A.wal 1393  =wceq 1395  e.wcel 1818  E!weu 2282  E*wmo 2283   cvv 3109
This theorem is referenced by:  tz7.44lem1  7090
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111
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