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Theorem csbiedf 3173
 Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 13-Oct-2016.)
Hypotheses
Ref Expression
csbiedf.1 xφ
csbiedf.2 (φxC)
csbiedf.3 (φA V)
csbiedf.4 ((φ x = A) → B = C)
Assertion
Ref Expression
csbiedf (φ[A / x]B = C)
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)   C(x)   V(x)

Proof of Theorem csbiedf
StepHypRef Expression
1 csbiedf.1 . . 3 xφ
2 csbiedf.4 . . . 4 ((φ x = A) → B = C)
32ex 423 . . 3 (φ → (x = AB = C))
41, 3alrimi 1765 . 2 (φx(x = AB = C))
5 csbiedf.3 . . 3 (φA V)
6 csbiedf.2 . . 3 (φxC)
7 csbiebt 3172 . . 3 ((A V xC) → (x(x = AB = C) ↔ [A / x]B = C))
85, 6, 7syl2anc 642 . 2 (φ → (x(x = AB = C) ↔ [A / x]B = C))
94, 8mpbid 201 1 (φ[A / x]B = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  Ⅎwnf 1544   = 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:  csbied  3178  csbie2t  3180
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