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Theorem eqsbc3r 2813
 Description: eqsbc3 2796 with setvar variable on right side of equals sign. (Contributed by Alan Sare, 24-Oct-2011.)
Assertion
Ref Expression
eqsbc3r (A B → ([A / x]𝐶 = x𝐶 = A))
Distinct variable groups:   x,𝐶   x,A
Allowed substitution hint:   B(x)

Proof of Theorem eqsbc3r
StepHypRef Expression
1 eqcom 2039 . . . . . 6 (𝐶 = xx = 𝐶)
21sbcbii 2812 . . . . 5 ([A / x]𝐶 = x[A / x]x = 𝐶)
32biimpi 113 . . . 4 ([A / x]𝐶 = x[A / x]x = 𝐶)
4 eqsbc3 2796 . . . 4 (A B → ([A / x]x = 𝐶A = 𝐶))
53, 4syl5ib 143 . . 3 (A B → ([A / x]𝐶 = xA = 𝐶))
6 eqcom 2039 . . 3 (A = 𝐶𝐶 = A)
75, 6syl6ib 150 . 2 (A B → ([A / x]𝐶 = x𝐶 = A))
8 idd 21 . . . . 5 (A B → (𝐶 = A𝐶 = A))
98, 6syl6ibr 151 . . . 4 (A B → (𝐶 = AA = 𝐶))
109, 4sylibrd 158 . . 3 (A B → (𝐶 = A[A / x]x = 𝐶))
1110, 2syl6ibr 151 . 2 (A B → (𝐶 = A[A / x]𝐶 = x))
127, 11impbid 120 1 (A B → ([A / x]𝐶 = x𝐶 = A))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsbc 2758 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759 This theorem is referenced by: (None)
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