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Theorem cbvoprab12 5578
 Description: Rule used to change first two bound variables in an operation abstraction, using implicit substitution. (Contributed by NM, 21-Feb-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.)
Hypotheses
Ref Expression
cbvoprab12.1
cbvoprab12.2
cbvoprab12.3
cbvoprab12.4
cbvoprab12.5
Assertion
Ref Expression
cbvoprab12
Distinct variable group:   ,,,,
Allowed substitution hints:   (,,,,)   (,,,,)

Proof of Theorem cbvoprab12
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . . 5
2 cbvoprab12.1 . . . . 5
31, 2nfan 1457 . . . 4
4 nfv 1421 . . . . 5
5 cbvoprab12.2 . . . . 5
64, 5nfan 1457 . . . 4
7 nfv 1421 . . . . 5
8 cbvoprab12.3 . . . . 5
97, 8nfan 1457 . . . 4
10 nfv 1421 . . . . 5
11 cbvoprab12.4 . . . . 5
1210, 11nfan 1457 . . . 4
13 opeq12 3551 . . . . . 6
1413eqeq2d 2051 . . . . 5
15 cbvoprab12.5 . . . . 5
1614, 15anbi12d 442 . . . 4
173, 6, 9, 12, 16cbvex2 1797 . . 3
1817opabbii 3824 . 2
19 dfoprab2 5552 . 2
20 dfoprab2 5552 . 2
2118, 19, 203eqtr4i 2070 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wnf 1349  wex 1381  cop 3378  copab 3817  coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-oprab 5516 This theorem is referenced by:  cbvoprab12v  5579  cbvmpt2x  5582  dfoprab4f  5819  fmpt2x  5826  tposoprab  5895
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