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Theorem sbcnestgf 3290
 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 3155 . . . . 5
2 csbeq1 3246 . . . . . 6
3 dfsbcq 3155 . . . . . 6
42, 3syl 16 . . . . 5
51, 4bibi12d 313 . . . 4
65imbi2d 308 . . 3
7 vex 2951 . . . . 5
87a1i 11 . . . 4
9 csbeq1a 3251 . . . . . 6
10 dfsbcq 3155 . . . . . 6
119, 10syl 16 . . . . 5
1211adantl 453 . . . 4
13 nfnf1 1808 . . . . 5
1413nfal 1864 . . . 4
15 nfa1 1806 . . . . 5
16 nfcsb1v 3275 . . . . . 6
1716a1i 11 . . . . 5
18 sp 1763 . . . . 5
1915, 17, 18nfsbcd 3173 . . . 4
208, 12, 14, 19sbciedf 3188 . . 3
216, 20vtoclg 3003 . 2
2221imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wal 1549  wnf 1553   wceq 1652   wcel 1725  wnfc 2558  cvv 2948  wsbc 3153  csb 3243 This theorem is referenced by:  csbnestgf  3291  sbcnestg  3292 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154  df-csb 3244
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