Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvoprab1 Structured version   Unicode version

Theorem cbvoprab1 6144
 Description: Rule used to change first bound variable in an operation abstraction, using implicit substitution. (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 5-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab1.1
cbvoprab1.2
cbvoprab1.3
Assertion
Ref Expression
cbvoprab1
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvoprab1
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1629 . . . . . 6
2 cbvoprab1.1 . . . . . 6
31, 2nfan 1846 . . . . 5
43nfex 1865 . . . 4
5 nfv 1629 . . . . . 6
6 cbvoprab1.2 . . . . . 6
75, 6nfan 1846 . . . . 5
87nfex 1865 . . . 4
9 opeq1 3984 . . . . . . 7
109eqeq2d 2447 . . . . . 6
11 cbvoprab1.3 . . . . . 6
1210, 11anbi12d 692 . . . . 5
1312exbidv 1636 . . . 4
144, 8, 13cbvex 1983 . . 3
1514opabbii 4272 . 2
16 dfoprab2 6121 . 2
17 dfoprab2 6121 . 2
1815, 16, 173eqtr4i 2466 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550  wnf 1553   wceq 1652  cop 3817  copab 4265  coprab 6082 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 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 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 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-opab 4267  df-oprab 6085
 Copyright terms: Public domain W3C validator