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Theorem csbief 3177
 Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 26-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbief.1 A V
csbief.2 xC
csbief.3 (x = AB = C)
Assertion
Ref Expression
csbief [A / x]B = C
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   C(x)

Proof of Theorem csbief
StepHypRef Expression
1 csbief.1 . 2 A V
2 csbief.2 . . . 4 xC
32a1i 10 . . 3 (A V → xC)
4 csbief.3 . . 3 (x = AB = C)
53, 4csbiegf 3176 . 2 (A V → [A / x]B = C)
61, 5ax-mp 8 1 [A / x]B = C
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [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:  csbing  3462  csbifg  3690  csbiotag  4371  csbopabg  4637  csbima12g  4955  csbovg  5552  eqerlem  5960
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