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Theorem sbcbr123 4452
 Description: Move substitution in and out of a binary relation. (Contributed by NM, 13-Dec-2005.) (Modified by NM, 22-Aug-2018.)
Assertion
Ref Expression
sbcbr123

Proof of Theorem sbcbr123
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3297 . . 3
2 csbeq1 3399 . . . 4
3 csbeq1 3399 . . . 4
4 csbeq1 3399 . . . 4
52, 3, 4breq123d 4415 . . 3
6 nfcsb1v 3412 . . . . 5
7 nfcsb1v 3412 . . . . 5
8 nfcsb1v 3412 . . . . 5
96, 7, 8nfbr 4445 . . . 4
10 csbeq1a 3405 . . . . 5
11 csbeq1a 3405 . . . . 5
12 csbeq1a 3405 . . . . 5
1310, 11, 12breq123d 4415 . . . 4
149, 13sbie 2110 . . 3
151, 5, 14vtoclbg 3137 . 2
16 sbcex 3304 . . . 4
1716con3i 135 . . 3
18 noel 3750 . . . . 5
19 df-br 4402 . . . . 5
2018, 19mtbir 299 . . . 4
21 csbprc 3782 . . . . 5
2221breqd 4412 . . . 4
2320, 22mtbiri 303 . . 3
2417, 232falsed 351 . 2
2515, 24pm2.61i 164 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wb 184   wceq 1370  wsb 1702   wcel 1758  cvv 3078  wsbc 3294  csb 3396  c0 3746  cop 3992   class class class wbr 4401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402 This theorem is referenced by:  sbcbr  4454  sbcbr12g  4455  csbcnvgALT  5133  sbcfung  5550  csbfv12  5835
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