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Theorem sbc2iegf 3112
 Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
sbc2iegf.1 xψ
sbc2iegf.2 yψ
sbc2iegf.3 x B W
sbc2iegf.4 ((x = A y = B) → (φψ))
Assertion
Ref Expression
sbc2iegf ((A V B W) → ([̣A / x]̣[̣B / yφψ))
Distinct variable groups:   x,y,A   y,B   x,V   y,W
Allowed substitution hints:   φ(x,y)   ψ(x,y)   B(x)   V(y)   W(x)

Proof of Theorem sbc2iegf
StepHypRef Expression
1 simpl 443 . 2 ((A V B W) → A V)
2 simpl 443 . . . 4 ((B W x = A) → B W)
3 sbc2iegf.4 . . . . 5 ((x = A y = B) → (φψ))
43adantll 694 . . . 4 (((B W x = A) y = B) → (φψ))
5 nfv 1619 . . . 4 y(B W x = A)
6 sbc2iegf.2 . . . . 5 yψ
76a1i 10 . . . 4 ((B W x = A) → Ⅎyψ)
82, 4, 5, 7sbciedf 3081 . . 3 ((B W x = A) → ([̣B / yφψ))
98adantll 694 . 2 (((A V B W) x = A) → ([̣B / yφψ))
10 nfv 1619 . . 3 x A V
11 sbc2iegf.3 . . 3 x B W
1210, 11nfan 1824 . 2 x(A V B W)
13 sbc2iegf.1 . . 3 xψ
1413a1i 10 . 2 ((A V B W) → Ⅎxψ)
151, 9, 12, 14sbciedf 3081 1 ((A V B W) → ([̣A / x]̣[̣B / yφψ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = 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:  sbc2ie  3113  opelopabaf  4710
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