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Theorem rmob 2844
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
rmoi.b
rmoi.c  C
Assertion
Ref Expression
rmob  C  C
Distinct variable groups:   ,   ,   , C   ,   ,
Allowed substitution hint:   ()

Proof of Theorem rmob
StepHypRef Expression
1 df-rmo 2308 . 2
2 simprl 483 . . . 4
3 eleq1 2097 . . . 4  C  C
42, 3syl5ibcom 144 . . 3  C  C
5 simpl 102 . . . 4  C  C
65a1i 9 . . 3  C  C
7 simplrl 487 . . . . 5  C
8 simpr 103 . . . . 5  C  C
9 simpll 481 . . . . 5  C
10 simplrr 488 . . . . 5  C
11 eleq1 2097 . . . . . . 7
12 rmoi.b . . . . . . 7
1311, 12anbi12d 442 . . . . . 6
14 eleq1 2097 . . . . . . 7  C  C
15 rmoi.c . . . . . . 7  C
1614, 15anbi12d 442 . . . . . 6  C  C
1713, 16mob 2717 . . . . 5  C  C  C
187, 8, 9, 7, 10, 17syl212anc 1144 . . . 4  C  C  C
1918ex 108 . . 3  C  C  C
204, 6, 19pm5.21ndd 620 . 2  C  C
211, 20sylanb 268 1  C  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242   wcel 1390  wmo 1898  wrmo 2303
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rmo 2308  df-v 2553
This theorem is referenced by:  rmoi  2845
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