 New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  csbie2g GIF version

Theorem csbie2g 3182
 Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3080 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)
Hypotheses
Ref Expression
csbie2g.1 (x = yB = C)
csbie2g.2 (y = AC = D)
Assertion
Ref Expression
csbie2g (A V[A / x]B = D)
Distinct variable groups:   x,y   y,A   y,B   x,C   y,D
Allowed substitution hints:   A(x)   B(x)   C(y)   D(x)   V(x,y)

Proof of Theorem csbie2g
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 df-csb 3137 . 2 [A / x]B = {z A / xz B}
2 csbie2g.1 . . . . 5 (x = yB = C)
32eleq2d 2420 . . . 4 (x = y → (z Bz C))
4 csbie2g.2 . . . . 5 (y = AC = D)
54eleq2d 2420 . . . 4 (y = A → (z Cz D))
63, 5sbcie2g 3079 . . 3 (A V → ([̣A / xz Bz D))
76abbi1dv 2469 . 2 (A V → {z A / xz B} = D)
81, 7syl5eq 2397 1 (A V[A / x]B = D)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136 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-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  df-csb 3137 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator