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Theorem sbcnestgf 2954
 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 2818 . . . . 5
2 csbeq1 2912 . . . . . 6
3 dfsbcq 2818 . . . . . 6
42, 3syl 14 . . . . 5
51, 4bibi12d 233 . . . 4
65imbi2d 228 . . 3
7 vex 2605 . . . . 5
87a1i 9 . . . 4
9 csbeq1a 2917 . . . . . 6
10 dfsbcq 2818 . . . . . 6
119, 10syl 14 . . . . 5
1211adantl 271 . . . 4
13 nfnf1 1477 . . . . 5
1413nfal 1509 . . . 4
15 nfa1 1475 . . . . 5
16 nfcsb1v 2939 . . . . . 6
1716a1i 9 . . . . 5
18 sp 1442 . . . . 5
1915, 17, 18nfsbcd 2835 . . . 4
208, 12, 14, 19sbciedf 2850 . . 3
216, 20vtoclg 2659 . 2
2221imp 122 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 102   wb 103  wal 1283   wceq 1285  wnf 1390   wcel 1434  wnfc 2207  cvv 2602  wsbc 2816  csb 2909 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-sbc 2817  df-csb 2910 This theorem is referenced by:  csbnestgf  2955  sbcnestg  2956
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