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Theorem csbiebg 3175
 Description: Bidirectional conversion between an implicit class substitution hypothesis x = A → B = C and its explicit substitution equivalent. (Contributed by NM, 24-Mar-2013.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
csbiebg.2 xC
Assertion
Ref Expression
csbiebg (A V → (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 csbiebg
Dummy variable a is distinct from all other variables.
StepHypRef Expression
1 eqeq2 2362 . . . 4 (a = A → (x = ax = A))
21imbi1d 308 . . 3 (a = A → ((x = aB = C) ↔ (x = AB = C)))
32albidv 1625 . 2 (a = A → (x(x = aB = C) ↔ x(x = AB = C)))
4 csbeq1 3139 . . 3 (a = A[a / x]B = [A / x]B)
54eqeq1d 2361 . 2 (a = A → ([a / x]B = C[A / x]B = C))
6 vex 2862 . . 3 a V
7 csbiebg.2 . . 3 xC
86, 7csbieb 3174 . 2 (x(x = aB = C) ↔ [a / x]B = C)
93, 5, 8vtoclbg 2915 1 (A V → (x(x = AB = C) ↔ [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-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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