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Theorem csbiebt 3172
 Description: Conversion of implicit substitution to explicit substitution into a class. (Closed theorem version of csbiegf 3176.) (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbiebt ((A V xC) → (x(x = AB = C) ↔ [A / x]B = C))
Distinct variable group:   x,A
Allowed substitution hints:   B(x)   C(x)   V(x)

Proof of Theorem csbiebt
StepHypRef Expression
1 elex 2867 . 2 (A VA V)
2 spsbc 3058 . . . . 5 (A V → (x(x = AB = C) → [̣A / x]̣(x = AB = C)))
32adantr 451 . . . 4 ((A V xC) → (x(x = AB = C) → [̣A / x]̣(x = AB = C)))
4 simpl 443 . . . . 5 ((A V xC) → A V)
5 biimt 325 . . . . . . 7 (x = A → (B = C ↔ (x = AB = C)))
6 csbeq1a 3144 . . . . . . . 8 (x = AB = [A / x]B)
76eqeq1d 2361 . . . . . . 7 (x = A → (B = C[A / x]B = C))
85, 7bitr3d 246 . . . . . 6 (x = A → ((x = AB = C) ↔ [A / x]B = C))
98adantl 452 . . . . 5 (((A V xC) x = A) → ((x = AB = C) ↔ [A / x]B = C))
10 nfv 1619 . . . . . 6 x A V
11 nfnfc1 2492 . . . . . 6 xxC
1210, 11nfan 1824 . . . . 5 x(A V xC)
13 nfcsb1v 3168 . . . . . . 7 x[A / x]B
1413a1i 10 . . . . . 6 ((A V xC) → x[A / x]B)
15 simpr 447 . . . . . 6 ((A V xC) → xC)
1614, 15nfeqd 2503 . . . . 5 ((A V xC) → Ⅎx[A / x]B = C)
174, 9, 12, 16sbciedf 3081 . . . 4 ((A V xC) → ([̣A / x]̣(x = AB = C) ↔ [A / x]B = C))
183, 17sylibd 205 . . 3 ((A V xC) → (x(x = AB = C) → [A / x]B = C))
1913a1i 10 . . . . . . . 8 (xCx[A / x]B)
20 id 19 . . . . . . . 8 (xCxC)
2119, 20nfeqd 2503 . . . . . . 7 (xC → Ⅎx[A / x]B = C)
2211, 21nfan1 1881 . . . . . 6 x(xC [A / x]B = C)
237biimprcd 216 . . . . . . 7 ([A / x]B = C → (x = AB = C))
2423adantl 452 . . . . . 6 ((xC [A / x]B = C) → (x = AB = C))
2522, 24alrimi 1765 . . . . 5 ((xC [A / x]B = C) → x(x = AB = C))
2625ex 423 . . . 4 (xC → ([A / x]B = Cx(x = AB = C)))
2726adantl 452 . . 3 ((A V xC) → ([A / x]B = Cx(x = AB = C)))
2818, 27impbid 183 . 2 ((A V xC) → (x(x = AB = C) ↔ [A / x]B = C))
291, 28sylan 457 1 ((A V xC) → (x(x = AB = C) ↔ [A / x]B = C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2476  Vcvv 2859  [̣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:  csbiedf  3173  csbieb  3174  csbiegf  3176
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