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Theorem cbvoprab12 6138
 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 1629 . . . . 5
2 cbvoprab12.1 . . . . 5
31, 2nfan 1846 . . . 4
4 nfv 1629 . . . . 5
5 cbvoprab12.2 . . . . 5
64, 5nfan 1846 . . . 4
7 nfv 1629 . . . . 5
8 cbvoprab12.3 . . . . 5
97, 8nfan 1846 . . . 4
10 nfv 1629 . . . . 5
11 cbvoprab12.4 . . . . 5
1210, 11nfan 1846 . . . 4
13 opeq12 3978 . . . . . 6
1413eqeq2d 2446 . . . . 5
15 cbvoprab12.5 . . . . 5
1614, 15anbi12d 692 . . . 4
173, 6, 9, 12, 16cbvex2 1991 . . 3
1817opabbii 4264 . 2
19 dfoprab2 6113 . 2
20 dfoprab2 6113 . 2
2118, 19, 203eqtr4i 2465 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550  wnf 1553   wceq 1652  cop 3809  copab 4257  coprab 6074 This theorem is referenced by:  cbvoprab12v  6139  cbvmpt2x  6142  dfoprab4f  6397  fmpt2x  6409  tposoprab  6507 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-opab 4259  df-oprab 6077
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