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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  F/  [.  ]. [.  ].  [.  ].

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  F/  [.  ]. [.  ].  [.  ].  F/  [.  ].
[.  ].  [.  ].
7 vex 2862 . . . . 5
87a1i 10 . . . 4  F/
9 csbeq1a 3144 . . . . . 6
10 dfsbcq 3048 . . . . . 6  [.  ].  [.  ].
119, 10syl 15 . . . . 5  [.  ].  [.  ].
1211adantl 452 . . . 4  F/  [.  ].  [.  ].
13 nfnf1 1790 . . . . 5  F/ F/
1413nfal 1842 . . . 4  F/ F/
15 nfa1 1788 . . . . 5  F/ F/
16 nfcsb1v 3168 . . . . . 6  F/_
1716a1i 10 . . . . 5  F/  F/_
18 sp 1747 . . . . 5  F/  F/
1915, 17, 18nfsbcd 3066 . . . 4  F/  F/ [.  ].
208, 12, 14, 19sbciedf 3081 . . 3  F/  [.  ].
[.  ].  [.  ].
216, 20vtoclg 2914 . 2  F/  [.  ]. [.  ].  [.  ].
2221imp 418 1  F/  [.  ]. [.  ].  [.  ].
Colors of variables: wff setvar class
Syntax hints:   wi 4   wb 176   wa 358  wal 1540   F/wnf 1544   wceq 1642   wcel 1710   F/_wnfc 2476  cvv 2859   [.wsbc 3046  csb 3136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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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