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

Theorem cbvriota 6562
 Description: Change bound variable in a restricted description binder. (Contributed by NM, 18-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypotheses
Ref Expression
cbvriota.1
cbvriota.2
cbvriota.3
Assertion
Ref Expression
cbvriota
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)

Proof of Theorem cbvriota
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cbvriota.1 . . . 4
2 cbvriota.2 . . . 4
3 cbvriota.3 . . . 4
41, 2, 3cbvreu 2932 . . 3
5 eleq1 2498 . . . . . 6
6 sbequ12 1945 . . . . . 6
75, 6anbi12d 693 . . . . 5
8 nfv 1630 . . . . 5
9 nfv 1630 . . . . . 6
10 nfs1v 2184 . . . . . 6
119, 10nfan 1847 . . . . 5
127, 8, 11cbviota 5425 . . . 4
13 eleq1 2498 . . . . . 6
14 sbequ 2113 . . . . . . 7
152, 3sbie 2151 . . . . . . 7
1614, 15syl6bb 254 . . . . . 6
1713, 16anbi12d 693 . . . . 5
18 nfv 1630 . . . . . 6
191nfsb 2187 . . . . . 6
2018, 19nfan 1847 . . . . 5
21 nfv 1630 . . . . 5
2217, 20, 21cbviota 5425 . . . 4
2312, 22eqtri 2458 . . 3
24 abid2 2555 . . . . 5
25 abid2 2555 . . . . 5
2624, 25eqtr4i 2461 . . . 4
2726fveq2i 5733 . . 3
284, 23, 27ifbieq12i 3762 . 2
29 df-riota 6551 . 2
30 df-riota 6551 . 2
3128, 29, 303eqtr4i 2468 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360  wnf 1554   wceq 1653  wsb 1659   wcel 1726  cab 2424  wreu 2709  cif 3741  cio 5418  cfv 5456  cund 6543  crio 6544 This theorem is referenced by:  cbvriotav  6563 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-iota 5420  df-fv 5464  df-riota 6551
 Copyright terms: Public domain W3C validator