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Theorem eqsbc3 2775
Description: Substitution applied to an atomic wff. Set theory version of eqsb3 2119. (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 2739 . 2 (y = A → ([y / x]x = B[A / x]x = B))
2 eqeq1 2024 . 2 (y = A → (y = BA = B))
3 sbsbc 2741 . . 3 ([y / x]x = B[y / x]x = B)
4 eqsb3 2119 . . 3 ([y / x]x = By = B)
53, 4bitr3i 175 . 2 ([y / x]x = By = B)
61, 2, 5vtoclbg 2587 1 (A 𝑉 → ([A / x]x = BA = B))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1226   wcel 1370  [wsb 1623  [wsbc 2737
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-sbc 2738
This theorem is referenced by:  sbceqal  2787  eqsbc3r  2792
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