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Theorem dfoprab4f 6405
 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 1629 . . . . 5
2 dfoprab4f.x . . . . . 6
3 nfs1v 2182 . . . . . 6
42, 3nfbi 1856 . . . . 5
51, 4nfim 1832 . . . 4
6 opeq1 3984 . . . . . 6
76eqeq2d 2447 . . . . 5
8 sbequ12 1944 . . . . . 6
98bibi2d 310 . . . . 5
107, 9imbi12d 312 . . . 4
11 nfv 1629 . . . . . 6
12 dfoprab4f.y . . . . . . 7
13 nfs1v 2182 . . . . . . 7
1412, 13nfbi 1856 . . . . . 6
1511, 14nfim 1832 . . . . 5
16 opeq2 3985 . . . . . . 7
1716eqeq2d 2447 . . . . . 6
18 sbequ12 1944 . . . . . . 7
1918bibi2d 310 . . . . . 6
2017, 19imbi12d 312 . . . . 5
21 dfoprab4f.1 . . . . 5
2215, 20, 21chvar 1968 . . . 4
235, 10, 22chvar 1968 . . 3
2423dfoprab4 6404 . 2
25 nfv 1629 . . 3
26 nfv 1629 . . 3
27 nfv 1629 . . . 4
2827, 3nfan 1846 . . 3
29 nfv 1629 . . . 4
3013nfsb 2185 . . . 4
3129, 30nfan 1846 . . 3
32 eleq1 2496 . . . . 5
33 eleq1 2496 . . . . 5
3432, 33bi2anan9 844 . . . 4
3518, 8sylan9bbr 682 . . . 4
3634, 35anbi12d 692 . . 3
3725, 26, 28, 31, 36cbvoprab12 6146 . 2
3824, 37eqtr4i 2459 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wnf 1553   wceq 1652  wsb 1658   wcel 1725  cop 3817  copab 4265   cxp 4876  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-13 1727  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-pow 4377  ax-pr 4403  ax-un 4701 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-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  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-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fv 5462  df-oprab 6085  df-1st 6349  df-2nd 6350
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