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Theorem sbcnestg 3185
 Description: Nest the composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Proof shortened by Mario Carneiro, 11-Nov-2016.)
Assertion
Ref Expression
sbcnestg (A V → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))
Distinct variable group:   φ,x
Allowed substitution hints:   φ(y)   A(x,y)   B(x,y)   V(x,y)

Proof of Theorem sbcnestg
StepHypRef Expression
1 nfv 1619 . . 3 xφ
21ax-gen 1546 . 2 yxφ
3 sbcnestgf 3183 . 2 ((A V yxφ) → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))
42, 3mpan2 652 1 (A V → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   ∈ wcel 1710  [̣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:  sbcco3g  3191
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