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Theorem csbiegf 3176
 Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 11-Nov-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiegf.1 (A VxC)
csbiegf.2 (x = AB = C)
Assertion
Ref Expression
csbiegf (A V[A / x]B = C)
Distinct variable groups:   x,A   x,V
Allowed substitution hints:   B(x)   C(x)

Proof of Theorem csbiegf
StepHypRef Expression
1 csbiegf.2 . . 3 (x = AB = C)
21ax-gen 1546 . 2 x(x = AB = C)
3 csbiegf.1 . . 3 (A VxC)
4 csbiebt 3172 . . 3 ((A V xC) → (x(x = AB = C) ↔ [A / x]B = C))
53, 4mpdan 649 . 2 (A V → (x(x = AB = C) ↔ [A / x]B = C))
62, 5mpbii 202 1 (A V[A / x]B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  [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:  csbief  3177  sbcco3g  3191  csbco3g  3193
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