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Theorem dfoprab4f 6858
 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:   ,,,   ,,,   ,,,   ,

Proof of Theorem dfoprab4f
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1707 . . . . 5
2 dfoprab4f.x . . . . . 6
3 nfs1v 2181 . . . . . 6
42, 3nfbi 1934 . . . . 5
51, 4nfim 1920 . . . 4
6 opeq1 4217 . . . . . 6
76eqeq2d 2471 . . . . 5
8 sbequ12 1992 . . . . . 6
98bibi2d 318 . . . . 5
107, 9imbi12d 320 . . . 4
11 nfv 1707 . . . . . 6
12 dfoprab4f.y . . . . . . 7
13 nfs1v 2181 . . . . . . 7
1412, 13nfbi 1934 . . . . . 6
1511, 14nfim 1920 . . . . 5
16 opeq2 4218 . . . . . . 7
1716eqeq2d 2471 . . . . . 6
18 sbequ12 1992 . . . . . . 7
1918bibi2d 318 . . . . . 6
2017, 19imbi12d 320 . . . . 5
21 dfoprab4f.1 . . . . 5
2215, 20, 21chvar 2013 . . . 4
235, 10, 22chvar 2013 . . 3
2423dfoprab4 6857 . 2
25 nfv 1707 . . 3
26 nfv 1707 . . 3
27 nfv 1707 . . . 4
2827, 3nfan 1928 . . 3
29 nfv 1707 . . . 4
3013nfsb 2184 . . . 4
3129, 30nfan 1928 . . 3
32 eleq1 2529 . . . . 5
33 eleq1 2529 . . . . 5
3432, 33bi2anan9 873 . . . 4
3518, 8sylan9bbr 700 . . . 4
3634, 35anbi12d 710 . . 3
3725, 26, 28, 31, 36cbvoprab12 6371 . 2
3824, 37eqtr4i 2489 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  F/wnf 1616  [wsb 1739  e.wcel 1818  <.cop 4035  {copab 4509  X.cxp 5002  {coprab 6297 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fv 5601  df-oprab 6300  df-1st 6800  df-2nd 6801
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