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Theorem sbc3ie 3115
 Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Jun-2014.) (Revised by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
sbc3ie.1 A V
sbc3ie.2 B V
sbc3ie.3 C V
sbc3ie.4 ((x = A y = B z = C) → (φψ))
Assertion
Ref Expression
sbc3ie ([̣A / x]̣[̣B / y]̣[̣C / zφψ)
Distinct variable groups:   x,y,z,A   y,B,z   z,C   ψ,x,y,z
Allowed substitution hints:   φ(x,y,z)   B(x)   C(x,y)

Proof of Theorem sbc3ie
StepHypRef Expression
1 sbc3ie.1 . 2 A V
2 sbc3ie.2 . 2 B V
3 sbc3ie.3 . . . 4 C V
43a1i 10 . . 3 ((x = A y = B) → C V)
5 sbc3ie.4 . . . 4 ((x = A y = B z = C) → (φψ))
653expa 1151 . . 3 (((x = A y = B) z = C) → (φψ))
74, 6sbcied 3082 . 2 ((x = A y = B) → ([̣C / zφψ))
81, 2, 7sbc2ie 3113 1 ([̣A / x]̣[̣B / y]̣[̣C / zφψ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934   = 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: (None)
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