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Theorem mgmidmo 15760
 Description: A two-sided identity element is unique (if it exists) in any magma. (Contributed by Mario Carneiro, 7-Dec-2014.) (Revised by NM, 17-Jun-2017.)
Assertion
Ref Expression
mgmidmo
Distinct variable groups:   ,,   , ,

Proof of Theorem mgmidmo
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 simpl 457 . . . . 5
21ralimi 2860 . . . 4
3 simpr 461 . . . . 5
43ralimi 2860 . . . 4
5 oveq1 6302 . . . . . . . . 9
6 id 22 . . . . . . . . 9
75, 6eqeq12d 2489 . . . . . . . 8
87rspcva 3217 . . . . . . 7
9 oveq2 6303 . . . . . . . . 9
10 id 22 . . . . . . . . 9
119, 10eqeq12d 2489 . . . . . . . 8
1211rspcva 3217 . . . . . . 7
138, 12sylan9req 2529 . . . . . 6
1413an42s 825 . . . . 5
1514ex 434 . . . 4
162, 4, 15syl2ani 656 . . 3
1716rgen2a 2894 . 2
18 oveq1 6302 . . . . . 6
1918eqeq1d 2469 . . . . 5
20 oveq2 6303 . . . . . 6
2120eqeq1d 2469 . . . . 5
2219, 21anbi12d 710 . . . 4
2322ralbidv 2906 . . 3
2423rmo4 3301 . 2
2517, 24mpbir 209 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1379   wcel 1767  wral 2817  wrmo 2820  (class class class)co 6295 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2822  df-rex 2823  df-rmo 2825  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-iota 5557  df-fv 5602  df-ov 6298 This theorem is referenced by:  ismgmid  15765  mndideu  15807
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