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Theorem eloprabga 5622
 Description: The law of concretion for operation class abstraction. Compare elopab 4021. (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:   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)   (,,)   (,,)

Proof of Theorem eloprabga
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 elex 2611 . 2
2 elex 2611 . 2
3 elex 2611 . 2
4 opexg 3991 . . . . 5
5 opexg 3991 . . . . 5
64, 5sylan 277 . . . 4
763impa 1134 . . 3
8 simpr 108 . . . . . . . . . . 11
98eqeq1d 2090 . . . . . . . . . 10
10 eqcom 2084 . . . . . . . . . . 11
11 vex 2605 . . . . . . . . . . . 12
12 vex 2605 . . . . . . . . . . . 12
13 vex 2605 . . . . . . . . . . . 12
1411, 12, 13otth2 4004 . . . . . . . . . . 11
1510, 14bitri 182 . . . . . . . . . 10
169, 15syl6bb 194 . . . . . . . . 9
1716anbi1d 453 . . . . . . . 8
18 eloprabga.1 . . . . . . . . 9
1918pm5.32i 442 . . . . . . . 8
2017, 19syl6bb 194 . . . . . . 7
21203exbidv 1791 . . . . . 6
22 df-oprab 5547 . . . . . . . . . 10
2322eleq2i 2146 . . . . . . . . 9
24 abid 2070 . . . . . . . . 9
2523, 24bitr2i 183 . . . . . . . 8
26 eleq1 2142 . . . . . . . 8
2725, 26syl5bb 190 . . . . . . 7
2827adantl 271 . . . . . 6
29 elisset 2614 . . . . . . . . . . 11
30 elisset 2614 . . . . . . . . . . 11
31 elisset 2614 . . . . . . . . . . 11
3229, 30, 313anim123i 1124 . . . . . . . . . 10
33 eeeanv 1850 . . . . . . . . . 10
3432, 33sylibr 132 . . . . . . . . 9
3534biantrurd 299 . . . . . . . 8
36 19.41vvv 1826 . . . . . . . 8
3735, 36syl6rbbr 197 . . . . . . 7
3837adantr 270 . . . . . 6
3921, 28, 383bitr3d 216 . . . . 5
4039expcom 114 . . . 4
4140vtocleg 2670 . . 3
427, 41mpcom 36 . 2
431, 2, 3, 42syl3an 1212 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103   w3a 920   wceq 1285  wex 1422   wcel 1434  cab 2068  cvv 2602  cop 3409  coprab 5544 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-oprab 5547 This theorem is referenced by:  eloprabg  5623  ovigg  5652
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