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Theorem moeq3 3013
 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 2362 . . . . . . 7
21anbi2d 684 . . . . . 6
3 biidd 228 . . . . . 6
4 biidd 228 . . . . . 6
52, 3, 43orbi123d 1251 . . . . 5
65eubidv 2212 . . . 4
7 vex 2862 . . . . 5
8 moeq3.1 . . . . 5
9 moeq3.2 . . . . 5
10 moeq3.3 . . . . 5
117, 8, 9, 10eueq3 3011 . . . 4
126, 11vtoclg 2914 . . 3
13 eumo 2244 . . 3
1412, 13syl 15 . 2
15 vex 2862 . . . . . . . . 9
16 eleq1 2413 . . . . . . . . 9
1715, 16mpbii 202 . . . . . . . 8
18 pm2.21 100 . . . . . . . 8
1917, 18syl5 28 . . . . . . 7
2019anim2d 548 . . . . . 6
2120orim1d 812 . . . . 5
22 3orass 937 . . . . 5
23 3orass 937 . . . . 5
2421, 22, 233imtr4g 261 . . . 4
2524alrimiv 1631 . . 3
26 euimmo 2253 . . 3
2725, 11, 26ee10 1376 . 2
2814, 27pm2.61i 156 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wo 357   wa 358   w3o 933  wal 1540   wceq 1642   wcel 1710  weu 2204  wmo 2205  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-nfc 2478  df-v 2861 This theorem is referenced by: (None)
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