Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfoprab4f Unicode version

Theorem dfoprab4f 5819
 Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
dfoprab4f.x
dfoprab4f.y
dfoprab4f.1
Assertion
Ref Expression
dfoprab4f
Distinct variable groups:   ,,,   ,,,   ,,,   ,
Allowed substitution hints:   (,,,)   (,,)   ()   ()

Proof of Theorem dfoprab4f
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1421 . . . . 5
2 dfoprab4f.x . . . . . 6
3 nfs1v 1815 . . . . . 6
42, 3nfbi 1481 . . . . 5
51, 4nfim 1464 . . . 4
6 opeq1 3549 . . . . . 6
76eqeq2d 2051 . . . . 5
8 sbequ12 1654 . . . . . 6
98bibi2d 221 . . . . 5
107, 9imbi12d 223 . . . 4
11 nfv 1421 . . . . . 6
12 dfoprab4f.y . . . . . . 7
13 nfs1v 1815 . . . . . . 7
1412, 13nfbi 1481 . . . . . 6
1511, 14nfim 1464 . . . . 5
16 opeq2 3550 . . . . . . 7
1716eqeq2d 2051 . . . . . 6
18 sbequ12 1654 . . . . . . 7
1918bibi2d 221 . . . . . 6
2017, 19imbi12d 223 . . . . 5
21 dfoprab4f.1 . . . . 5
2215, 20, 21chvar 1640 . . . 4
235, 10, 22chvar 1640 . . 3
2423dfoprab4 5818 . 2
25 nfv 1421 . . 3
26 nfv 1421 . . 3
27 nfv 1421 . . . 4
2827, 3nfan 1457 . . 3
29 nfv 1421 . . . 4
3013nfsb 1822 . . . 4
3129, 30nfan 1457 . . 3
32 eleq1 2100 . . . . 5
33 eleq1 2100 . . . . 5
3432, 33bi2anan9 538 . . . 4
3518, 8sylan9bbr 436 . . . 4
3634, 35anbi12d 442 . . 3
3725, 26, 28, 31, 36cbvoprab12 5578 . 2
3824, 37eqtr4i 2063 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wnf 1349   wcel 1393  wsb 1645  cop 3378  copab 3817   cxp 4343  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-13 1404  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  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-oprab 5516  df-1st 5767  df-2nd 5768 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator