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Theorem sbcnestgf 3839
Description: Nest the composition of two substitutions. (Contributed by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestgf

Proof of Theorem sbcnestgf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3329 . . . . 5
2 csbeq1 3437 . . . . . 6
32sbceq1d 3332 . . . . 5
41, 3bibi12d 321 . . . 4
54imbi2d 316 . . 3
6 vex 3112 . . . . 5
76a1i 11 . . . 4
8 csbeq1a 3443 . . . . . 6
98sbceq1d 3332 . . . . 5
109adantl 466 . . . 4
11 nfnf1 1899 . . . . 5
1211nfal 1947 . . . 4
13 nfa1 1897 . . . . 5
14 nfcsb1v 3450 . . . . . 6
1514a1i 11 . . . . 5
16 sp 1859 . . . . 5
1713, 15, 16nfsbcd 3348 . . . 4
187, 10, 12, 17sbciedf 3363 . . 3
195, 18vtoclg 3167 . 2
2019imp 429 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  <->wb 184  /\wa 369  A.wal 1393  =wceq 1395  F/wnf 1616  e.wcel 1818  F/_wnfc 2605   cvv 3109  [.wsbc 3327  [_csb 3434
This theorem is referenced by:  csbnestgf  3840  sbcnestg  3841
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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3435
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