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Theorem eloprabga 6152
 Description: The law of concretion for operation class abstraction. Compare elopab 4454. (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 2956 . 2
2 elex 2956 . 2
3 elex 2956 . 2
4 opex 4419 . . 3
5 simpr 448 . . . . . . . . . 10
65eqeq1d 2443 . . . . . . . . 9
7 eqcom 2437 . . . . . . . . . 10
8 vex 2951 . . . . . . . . . . 11
9 vex 2951 . . . . . . . . . . 11
10 vex 2951 . . . . . . . . . . 11
118, 9, 10otth2 4431 . . . . . . . . . 10
127, 11bitri 241 . . . . . . . . 9
136, 12syl6bb 253 . . . . . . . 8
1413anbi1d 686 . . . . . . 7
15 eloprabga.1 . . . . . . . 8
1615pm5.32i 619 . . . . . . 7
1714, 16syl6bb 253 . . . . . 6
18173exbidv 1639 . . . . 5
19 df-oprab 6077 . . . . . . . . 9
2019eleq2i 2499 . . . . . . . 8
21 abid 2423 . . . . . . . 8
2220, 21bitr2i 242 . . . . . . 7
23 eleq1 2495 . . . . . . 7
2422, 23syl5bb 249 . . . . . 6
2524adantl 453 . . . . 5
26 elisset 2958 . . . . . . . . . 10
27 elisset 2958 . . . . . . . . . 10
28 elisset 2958 . . . . . . . . . 10
2926, 27, 283anim123i 1139 . . . . . . . . 9
30 eeeanv 1938 . . . . . . . . 9
3129, 30sylibr 204 . . . . . . . 8
3231biantrurd 495 . . . . . . 7
33 19.41vvv 1926 . . . . . . 7
3432, 33syl6rbbr 256 . . . . . 6
3534adantr 452 . . . . 5
3618, 25, 353bitr3d 275 . . . 4
3736expcom 425 . . 3
384, 37vtocle 3017 . 2
391, 2, 3, 38syl3an 1226 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   w3a 936  wex 1550   wceq 1652   wcel 1725  cab 2421  cvv 2948  cop 3809  coprab 6074 This theorem is referenced by:  eloprabg  6153  ovigg  6186  vdwpc  13340 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-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 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-ne 2600  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-oprab 6077
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