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Theorem toycom 29707
 Description: Show the commutative law for an operation on a toy structure class of commuatitive operations on . This illustrates how a structure class can be partially specialized. In practice, we would ordinarily define a new constant such as "CAbel" in place of . (Contributed by NM, 17-Mar-2013.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
toycom.1
toycom.2
Assertion
Ref Expression
toycom
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()

Proof of Theorem toycom
StepHypRef Expression
1 toycom.1 . . . . . 6
2 ssrab2 3420 . . . . . 6
31, 2eqsstri 3370 . . . . 5
43sseli 3336 . . . 4
6 simp2 958 . . . 4
7 fveq2 5720 . . . . . . . 8
87eqeq1d 2443 . . . . . . 7
98, 1elrab2 3086 . . . . . 6
109simprbi 451 . . . . 5
11103ad2ant1 978 . . . 4
126, 11eleqtrrd 2512 . . 3
13 simp3 959 . . . 4
1413, 11eleqtrrd 2512 . . 3
15 eqid 2435 . . . 4
16 eqid 2435 . . . 4
1715, 16ablcom 15421 . . 3
185, 12, 14, 17syl3anc 1184 . 2
19 toycom.2 . . 3
2019oveqi 6086 . 2
2119oveqi 6086 . 2
2218, 20, 213eqtr4g 2492 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 936   wceq 1652   wcel 1725  crab 2701  cfv 5446  (class class class)co 6073  cc 8980  cbs 13461   cplusg 13521  cabel 15405 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-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454  df-ov 6076  df-cmn 15406  df-abl 15407
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