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Theorem sbcss 3660
 Description: Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcss (A B → ([̣A / xC D[A / x]C [A / x]D))

Proof of Theorem sbcss
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 sbcalg 3094 . . 3 (A B → ([̣A / xy(y Cy D) ↔ yA / x]̣(y Cy D)))
2 sbcimg 3087 . . . . 5 (A B → ([̣A / x]̣(y Cy D) ↔ ([̣A / xy C → [̣A / xy D)))
3 sbcel2g 3157 . . . . . 6 (A B → ([̣A / xy Cy [A / x]C))
4 sbcel2g 3157 . . . . . 6 (A B → ([̣A / xy Dy [A / x]D))
53, 4imbi12d 311 . . . . 5 (A B → (([̣A / xy C → [̣A / xy D) ↔ (y [A / x]Cy [A / x]D)))
62, 5bitrd 244 . . . 4 (A B → ([̣A / x]̣(y Cy D) ↔ (y [A / x]Cy [A / x]D)))
76albidv 1625 . . 3 (A B → (yA / x]̣(y Cy D) ↔ y(y [A / x]Cy [A / x]D)))
81, 7bitrd 244 . 2 (A B → ([̣A / xy(y Cy D) ↔ y(y [A / x]Cy [A / x]D)))
9 dfss2 3262 . . 3 (C Dy(y Cy D))
109sbcbii 3101 . 2 ([̣A / xC D ↔ [̣A / xy(y Cy D))
11 dfss2 3262 . 2 ([A / x]C [A / x]Dy(y [A / x]Cy [A / x]D))
128, 10, 113bitr4g 279 1 (A B → ([̣A / xC D[A / x]C [A / x]D))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710  [̣wsbc 3046  [csb 3136   ⊆ wss 3257 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-nan 1288  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  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by: (None)
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