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Theorem eqsbc3 3085
 Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2454. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
eqsbc3 (A V → ([̣A / xx = 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 3048 . 2 (y = A → ([̣y / xx = B ↔ [̣A / xx = B))
2 eqeq1 2359 . 2 (y = A → (y = BA = B))
3 sbsbc 3050 . . 3 ([y / x]x = B ↔ [̣y / xx = B)
4 eqsb3 2454 . . 3 ([y / x]x = By = B)
53, 4bitr3i 242 . 2 ([̣y / xx = By = B)
61, 2, 5vtoclbg 2915 1 (A V → ([̣A / xx = BA = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  [̣wsbc 3046 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-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:  sbceqal  3097  eqsbc3r  3103
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