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Theorem sbc2ie 3113
 Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2ie.1 A V
sbc2ie.2 B V
sbc2ie.3 ((x = A y = B) → (φψ))
Assertion
Ref Expression
sbc2ie ([̣A / x]̣[̣B / yφψ)
Distinct variable groups:   x,y,A   y,B   ψ,x,y
Allowed substitution hints:   φ(x,y)   B(x)

Proof of Theorem sbc2ie
StepHypRef Expression
1 sbc2ie.1 . 2 A V
2 sbc2ie.2 . 2 B V
3 nfv 1619 . . 3 xψ
4 nfv 1619 . . 3 yψ
52nfth 1553 . . 3 x B V
6 sbc2ie.3 . . 3 ((x = A y = B) → (φψ))
73, 4, 5, 6sbc2iegf 3112 . 2 ((A V B V) → ([̣A / x]̣[̣B / yφψ))
81, 2, 7mp2an 653 1 ([̣A / x]̣[̣B / yφψ)
 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-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:  sbc3ie  3115
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