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Theorem csboprabg 31801
 Description: Move class substitution in and out of class abstractions of nested ordered pairs. (Contributed by ML, 25-Oct-2020.)
Assertion
Ref Expression
csboprabg
Distinct variable groups:   ,   ,   ,   ,   ,   ,   ,   ,   ,
Allowed substitution hints:   (,,,)   ()   ()

Proof of Theorem csboprabg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbab 3801 . . 3
2 sbcex2 3306 . . . . 5
3 sbcex2 3306 . . . . . . 7
4 sbcex2 3306 . . . . . . . . 9
5 sbcan 3298 . . . . . . . . . . 11
6 sbcg 3321 . . . . . . . . . . . 12
76anbi1d 719 . . . . . . . . . . 11
85, 7syl5bb 265 . . . . . . . . . 10
98exbidv 1776 . . . . . . . . 9
104, 9syl5bb 265 . . . . . . . 8
1110exbidv 1776 . . . . . . 7
123, 11syl5bb 265 . . . . . 6
1312exbidv 1776 . . . . 5
142, 13syl5bb 265 . . . 4
1514abbidv 2589 . . 3
161, 15syl5eq 2517 . 2
17 df-oprab 6312 . . 3
1817csbeq2i 3786 . 2
19 df-oprab 6312 . 2
2016, 18, 193eqtr4g 2530 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376   wceq 1452  wex 1671   wcel 1904  cab 2457  wsbc 3255  csb 3349  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-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-oprab 6312 This theorem is referenced by:  csbmpt22g  31802
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