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Theorem csbie2 3181
 Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by NM, 27-Aug-2007.)
Hypotheses
Ref Expression
csbie2t.1 A V
csbie2t.2 B V
csbie2.3 ((x = A y = B) → C = D)
Assertion
Ref Expression
csbie2 [A / x][B / y]C = D
Distinct variable groups:   x,y,A   x,B,y   x,D,y
Allowed substitution hints:   C(x,y)

Proof of Theorem csbie2
StepHypRef Expression
1 csbie2.3 . . 3 ((x = A y = B) → C = D)
21gen2 1547 . 2 xy((x = A y = B) → C = D)
3 csbie2t.1 . . 3 A V
4 csbie2t.2 . . 3 B V
53, 4csbie2t 3180 . 2 (xy((x = A y = B) → C = D) → [A / x][B / y]C = D)
62, 5ax-mp 8 1 [A / x][B / y]C = D
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  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: (None)
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