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Theorem eqsbc3 2796
 Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2138. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
eqsbc3 (A 𝑉 → ([A / x]x = BA = B))
Distinct variable group:   x,B
Allowed substitution hints:   A(x)   𝑉(x)

Proof of Theorem eqsbc3
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 2760 . 2 (y = A → ([y / x]x = B[A / x]x = B))
2 eqeq1 2043 . 2 (y = A → (y = BA = B))
3 sbsbc 2762 . . 3 ([y / x]x = B[y / x]x = B)
4 eqsb3 2138 . . 3 ([y / x]x = By = B)
53, 4bitr3i 175 . 2 ([y / x]x = By = B)
61, 2, 5vtoclbg 2608 1 (A 𝑉 → ([A / x]x = BA = B))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   = wceq 1242   ∈ wcel 1390  [wsb 1642  [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:  sbceqal  2808  eqsbc3r  2813
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