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Theorem exidu1 21906
 Description: Unicity of the left and right identity element of a magma when it exists. (Contributed by FL, 12-Dec-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)
Hypothesis
Ref Expression
exidu1.1
Assertion
Ref Expression
exidu1
Distinct variable groups:   ,,   ,,

Proof of Theorem exidu1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 exidu1.1 . . 3
21isexid2 21905 . 2
3 simpl 444 . . . . . . . 8
43ralimi 2773 . . . . . . 7
5 oveq2 6081 . . . . . . . . 9
6 id 20 . . . . . . . . 9
75, 6eqeq12d 2449 . . . . . . . 8
87rspcv 3040 . . . . . . 7
94, 8syl5 30 . . . . . 6
10 simpr 448 . . . . . . . 8
1110ralimi 2773 . . . . . . 7
12 oveq1 6080 . . . . . . . . 9
13 id 20 . . . . . . . . 9
1412, 13eqeq12d 2449 . . . . . . . 8
1514rspcv 3040 . . . . . . 7
1611, 15syl5 30 . . . . . 6
179, 16im2anan9r 810 . . . . 5
18 eqtr2 2453 . . . . . 6
1918eqcomd 2440 . . . . 5
2017, 19syl6 31 . . . 4
2120rgen2a 2764 . . 3
2221a1i 11 . 2
23 oveq1 6080 . . . . . 6
2423eqeq1d 2443 . . . . 5
25 oveq2 6081 . . . . . 6
2625eqeq1d 2443 . . . . 5
2724, 26anbi12d 692 . . . 4
2827ralbidv 2717 . . 3
2928reu4 3120 . 2
302, 22, 29sylanbrc 646 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2697  wrex 2698  wreu 2699   cin 3311   crn 4871  (class class class)co 6073   cexid 21894  cmagm 21898 This theorem is referenced by:  iorlid  21908  cmpidelt  21909  exidresid  26545 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 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-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  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-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-ov 6076  df-exid 21895  df-mgm 21899
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