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Theorem moeq3 3233
 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 2466 . . . . . . 7
21anbi2d 703 . . . . . 6
3 biidd 237 . . . . . 6
4 biidd 237 . . . . . 6
52, 3, 43orbi123d 1289 . . . . 5
65eubidv 2283 . . . 4
7 vex 3071 . . . . 5
8 moeq3.1 . . . . 5
9 moeq3.2 . . . . 5
10 moeq3.3 . . . . 5
117, 8, 9, 10eueq3 3231 . . . 4
126, 11vtoclg 3126 . . 3
13 eumo 2293 . . 3
1412, 13syl 16 . 2
15 eqvisset 3076 . . . . . . . 8
16 pm2.21 108 . . . . . . . 8
1715, 16syl5 32 . . . . . . 7
1817anim2d 565 . . . . . 6
1918orim1d 835 . . . . 5
20 3orass 968 . . . . 5
21 3orass 968 . . . . 5
2219, 20, 213imtr4g 270 . . . 4
2322alrimiv 1686 . . 3
24 euimmo 2330 . . 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 964  wal 1368   wceq 1370   wcel 1758  weu 2260  wmo 2261  cvv 3068 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-v 3070 This theorem is referenced by:  tz7.44lem1  6961
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