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Theorem mob 3108
 Description: Equality implied by "at most one." (Contributed by NM, 18-Feb-2006.)
Hypotheses
Ref Expression
moi.1
moi.2
Assertion
Ref Expression
mob
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem mob
StepHypRef Expression
1 elex 2956 . . . . 5
2 nfcv 2571 . . . . . . . 8
3 nfv 1629 . . . . . . . . . 10
4 nfmo1 2291 . . . . . . . . . 10
5 nfv 1629 . . . . . . . . . 10
63, 4, 5nf3an 1849 . . . . . . . . 9
7 nfv 1629 . . . . . . . . 9
86, 7nfim 1832 . . . . . . . 8
9 moi.1 . . . . . . . . . 10
1093anbi3d 1260 . . . . . . . . 9
11 eqeq1 2441 . . . . . . . . . 10
1211bibi1d 311 . . . . . . . . 9
1310, 12imbi12d 312 . . . . . . . 8
14 moi.2 . . . . . . . . 9
1514mob2 3106 . . . . . . . 8
162, 8, 13, 15vtoclgf 3002 . . . . . . 7
1716com12 29 . . . . . 6
18173expib 1156 . . . . 5
191, 18syl 16 . . . 4
2019com3r 75 . . 3
2120imp 419 . 2
22213impib 1151 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936   wceq 1652   wcel 1725  wmo 2281  cvv 2948 This theorem is referenced by:  moi  3109  rmob  3241 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 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950
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