Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  sbcco3g Structured version   Unicode version

Theorem sbcco3g 3818
 Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1
Assertion
Ref Expression
sbcco3g
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   (,)   ()   (,)

Proof of Theorem sbcco3g
StepHypRef Expression
1 sbcnestg 3816 . 2
2 elex 3089 . . 3
3 nfcvd 2581 . . . 4
4 sbcco3g.1 . . . 4
53, 4csbiegf 3419 . . 3
6 dfsbcq 3301 . . 3
72, 5, 63syl 18 . 2
81, 7bitrd 256 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 187   wceq 1437   wcel 1872  cvv 3080  wsbc 3299  csb 3395 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401 This theorem depends on definitions:  df-bi 188  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-sbc 3300  df-csb 3396 This theorem is referenced by:  fzshftral  11890  2rexfrabdioph  35609  3rexfrabdioph  35610  4rexfrabdioph  35611  6rexfrabdioph  35612  7rexfrabdioph  35613
 Copyright terms: Public domain W3C validator