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Theorem csbin 3803
 Description: Distribute proper substitution into a class through an intersection relation. (Contributed by Alan Sare, 22-Jul-2012.) (Revised by NM, 18-Aug-2018.)
Assertion
Ref Expression
csbin

Proof of Theorem csbin
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3352 . . . 4
2 csbeq1 3352 . . . . 5
3 csbeq1 3352 . . . . 5
42, 3ineq12d 3626 . . . 4
51, 4eqeq12d 2486 . . 3
6 vex 3034 . . . 4
7 nfcsb1v 3365 . . . . 5
8 nfcsb1v 3365 . . . . 5
97, 8nfin 3630 . . . 4
10 csbeq1a 3358 . . . . 5
11 csbeq1a 3358 . . . . 5
1210, 11ineq12d 3626 . . . 4
136, 9, 12csbief 3374 . . 3
145, 13vtoclg 3093 . 2
15 csbprc 3774 . . 3
16 csbprc 3774 . . . . 5
17 csbprc 3774 . . . . 5
1816, 17ineq12d 3626 . . . 4
19 in0 3763 . . . 4
2018, 19syl6req 2522 . . 3
2115, 20eqtrd 2505 . 2
2214, 21pm2.61i 169 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wceq 1452   wcel 1904  cvv 3031  csb 3349   cin 3389  c0 3722 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-3an 1009  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 This theorem is referenced by:  csbres  5114  disjxpin  28275  csbpredg  31797  onfrALTlem5  36978  onfrALTlem4  36979  disjinfi  37539
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