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Theorem sbcied2 3083
 Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by NM, 13-Dec-2014.)
Hypotheses
Ref Expression
sbcied2.1 (φA V)
sbcied2.2 (φA = B)
sbcied2.3 ((φ x = B) → (ψχ))
Assertion
Ref Expression
sbcied2 (φ → ([̣A / xψχ))
Distinct variable groups:   x,A   φ,x   χ,x
Allowed substitution hints:   ψ(x)   B(x)   V(x)

Proof of Theorem sbcied2
StepHypRef Expression
1 sbcied2.1 . 2 (φA V)
2 id 19 . . . 4 (x = Ax = A)
3 sbcied2.2 . . . 4 (φA = B)
42, 3sylan9eqr 2407 . . 3 ((φ x = A) → x = B)
5 sbcied2.3 . . 3 ((φ x = B) → (ψχ))
64, 5syldan 456 . 2 ((φ x = A) → (ψχ))
71, 6sbcied 3082 1 (φ → ([̣A / xψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  [̣wsbc 3046 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 This theorem is referenced by: (None)
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