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Theorem sbcco3g 3191
 Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (Revised by Mario Carneiro, 11-Nov-2016.)
Hypothesis
Ref Expression
sbcco3g.1 (x = AB = C)
Assertion
Ref Expression
sbcco3g (A V → ([̣A / x]̣[̣B / yφ ↔ [̣C / yφ))
Distinct variable groups:   x,A   φ,x   x,C
Allowed substitution hints:   φ(y)   A(y)   B(x,y)   C(y)   V(x,y)

Proof of Theorem sbcco3g
StepHypRef Expression
1 sbcnestg 3185 . 2 (A V → ([̣A / x]̣[̣B / yφ ↔ [̣[A / x]B / yφ))
2 elex 2867 . . 3 (A VA V)
3 nfcvd 2490 . . . 4 (A V → xC)
4 sbcco3g.1 . . . 4 (x = AB = C)
53, 4csbiegf 3176 . . 3 (A V → [A / x]B = C)
6 dfsbcq 3048 . . 3 ([A / x]B = C → ([̣[A / x]B / yφ ↔ [̣C / yφ))
72, 5, 63syl 18 . 2 (A V → ([̣[A / x]B / yφ ↔ [̣C / yφ))
81, 7bitrd 244 1 (A V → ([̣A / x]̣[̣B / yφ ↔ [̣C / yφ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  Vcvv 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:  sbcco3gOLD  3192
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