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Theorem sbc2iedv 3114
 Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Proof shortened by Mario Carneiro, 18-Oct-2016.)
Hypotheses
Ref Expression
sbc2iedv.1 A V
sbc2iedv.2 B V
sbc2iedv.3 (φ → ((x = A y = B) → (ψχ)))
Assertion
Ref Expression
sbc2iedv (φ → ([̣A / x]̣[̣B / yψχ))
Distinct variable groups:   x,y,A   y,B   φ,x,y   χ,x,y
Allowed substitution hints:   ψ(x,y)   B(x)

Proof of Theorem sbc2iedv
StepHypRef Expression
1 sbc2iedv.1 . . 3 A V
21a1i 10 . 2 (φA V)
3 sbc2iedv.2 . . . 4 B V
43a1i 10 . . 3 ((φ x = A) → B V)
5 sbc2iedv.3 . . . 4 (φ → ((x = A y = B) → (ψχ)))
65impl 603 . . 3 (((φ x = A) y = B) → (ψχ))
74, 6sbcied 3082 . 2 ((φ x = A) → ([̣B / yψχ))
82, 7sbcied 3082 1 (φ → ([̣A / x]̣[̣B / yψχ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  Vcvv 2859  [̣wsbc 3046 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 This theorem is referenced by: (None)
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