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Theorem eloprabga 6389
Description: The law of concretion for operation class abstraction. Compare elopab 4760. (Contributed by NM, 14-Sep-1999.) (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
eloprabga.1
Assertion
Ref Expression
eloprabga
Distinct variable groups:   , , ,   , , ,   , , ,   , , ,

Proof of Theorem eloprabga
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 3118 . 2
2 elex 3118 . 2
3 elex 3118 . 2
4 opex 4716 . . 3
5 simpr 461 . . . . . . . . . 10
65eqeq1d 2459 . . . . . . . . 9
7 eqcom 2466 . . . . . . . . . 10
8 vex 3112 . . . . . . . . . . 11
9 vex 3112 . . . . . . . . . . 11
10 vex 3112 . . . . . . . . . . 11
118, 9, 10otth2 4733 . . . . . . . . . 10
127, 11bitri 249 . . . . . . . . 9
136, 12syl6bb 261 . . . . . . . 8
1413anbi1d 704 . . . . . . 7
15 eloprabga.1 . . . . . . . 8
1615pm5.32i 637 . . . . . . 7
1714, 16syl6bb 261 . . . . . 6
18173exbidv 1717 . . . . 5
19 df-oprab 6300 . . . . . . . . 9
2019eleq2i 2535 . . . . . . . 8
21 abid 2444 . . . . . . . 8
2220, 21bitr2i 250 . . . . . . 7
23 eleq1 2529 . . . . . . 7
2422, 23syl5bb 257 . . . . . 6
2524adantl 466 . . . . 5
26 elisset 3120 . . . . . . . . . 10
27 elisset 3120 . . . . . . . . . 10
28 elisset 3120 . . . . . . . . . 10
2926, 27, 283anim123i 1181 . . . . . . . . 9
30 eeeanv 1989 . . . . . . . . 9
3129, 30sylibr 212 . . . . . . . 8
3231biantrurd 508 . . . . . . 7
33 19.41vvv 1773 . . . . . . 7
3432, 33syl6rbbr 264 . . . . . 6
3534adantr 465 . . . . 5
3618, 25, 353bitr3d 283 . . . 4
3736expcom 435 . . 3
384, 37vtocle 3183 . 2
391, 2, 3, 38syl3an 1270 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  /\w3a 973  =wceq 1395  E.wex 1612  e.wcel 1818  {cab 2442   cvv 3109  <.cop 4035  {coprab 6297
This theorem is referenced by:  eloprabg  6390  ovigg  6423  vdwpc  14498  isrgra  24926  isrusgra  24927  elmpps  28933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-oprab 6300
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