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Theorem sbcnestgf 3183
 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 3048 . . . . 5
2 csbeq1 3139 . . . . . 6
3 dfsbcq 3048 . . . . . 6
42, 3syl 15 . . . . 5
51, 4bibi12d 312 . . . 4
65imbi2d 307 . . 3
7 vex 2862 . . . . 5
87a1i 10 . . . 4
9 csbeq1a 3144 . . . . . 6
10 dfsbcq 3048 . . . . . 6
119, 10syl 15 . . . . 5
1211adantl 452 . . . 4
13 nfnf1 1790 . . . . 5
1413nfal 1842 . . . 4
15 nfa1 1788 . . . . 5
16 nfcsb1v 3168 . . . . . 6
1716a1i 10 . . . . 5
18 sp 1747 . . . . 5
1915, 17, 18nfsbcd 3066 . . . 4
208, 12, 14, 19sbciedf 3081 . . 3
216, 20vtoclg 2914 . 2
2221imp 418 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 176   wa 358  wal 1540  wnf 1544   wceq 1642   wcel 1710  wnfc 2476  cvv 2859  wsbc 3046  csb 3136 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137 This theorem is referenced by:  csbnestgf  3184  sbcnestg  3185
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