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Theorem eqsbc3r 3103
 Description: eqsbc3 3085 with setvar variable on right side of equals sign. This proof was automatically generated from the virtual deduction proof eqsbc3rVD in set.mm using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
eqsbc3r (A B → ([̣A / xC = xC = A))
Distinct variable groups:   x,C   x,A
Allowed substitution hint:   B(x)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqcom 2355 . . . . . 6 (C = xx = C)
21sbcbii 3101 . . . . 5 ([̣A / xC = x ↔ [̣A / xx = C)
32biimpi 186 . . . 4 ([̣A / xC = x → [̣A / xx = C)
4 eqsbc3 3085 . . . 4 (A B → ([̣A / xx = CA = C))
53, 4syl5ib 210 . . 3 (A B → ([̣A / xC = xA = C))
6 eqcom 2355 . . 3 (A = CC = A)
75, 6syl6ib 217 . 2 (A B → ([̣A / xC = xC = A))
8 idd 21 . . . . 5 (A B → (C = AC = A))
98, 6syl6ibr 218 . . . 4 (A B → (C = AA = C))
109, 4sylibrd 225 . . 3 (A B → (C = A → [̣A / xx = C))
1110, 2syl6ibr 218 . 2 (A B → (C = A → [̣A / xC = x))
127, 11impbid 183 1 (A B → ([̣A / xC = xC = A))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = 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-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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