Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cbvoprab2 Structured version   Visualization version   Unicode version

Theorem cbvoprab2 6383
 Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypotheses
Ref Expression
cbvoprab2.1
cbvoprab2.2
cbvoprab2.3
Assertion
Ref Expression
cbvoprab2
Distinct variable group:   ,,,
Allowed substitution hints:   (,,,)   (,,,)

Proof of Theorem cbvoprab2
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfv 1769 . . . . . . 7
2 cbvoprab2.1 . . . . . . 7
31, 2nfan 2031 . . . . . 6
43nfex 2050 . . . . 5
5 nfv 1769 . . . . . . 7
6 cbvoprab2.2 . . . . . . 7
75, 6nfan 2031 . . . . . 6
87nfex 2050 . . . . 5
9 opeq2 4159 . . . . . . . . 9
109opeq1d 4164 . . . . . . . 8
1110eqeq2d 2481 . . . . . . 7
12 cbvoprab2.3 . . . . . . 7
1311, 12anbi12d 725 . . . . . 6
1413exbidv 1776 . . . . 5
154, 8, 14cbvex 2128 . . . 4
1615exbii 1726 . . 3
1716abbii 2587 . 2
18 df-oprab 6312 . 2
19 df-oprab 6312 . 2
2017, 18, 193eqtr4i 2503 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452  wex 1671  wnf 1675  cab 2457  cop 3965  coprab 6309 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-oprab 6312 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator