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Theorem eqsbc1 3086
Description: Substitution for the left-hand side in an equality. Class version of eqsb1 2454. (Contributed by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
eqsbc1 (A V → ([̣A / xx = BA = B))
Distinct variable group:   x,B
Allowed substitution hints:   A(x)   V(x)

Proof of Theorem eqsbc1
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfsbcq 3049 . 2 (y = A → ([̣y / xx = B ↔ [̣A / xx = B))
2 eqeq1 2359 . 2 (y = A → (y = BA = B))
3 sbsbc 3051 . . 3 ([y / x]x = B ↔ [̣y / xx = B)
4 eqsb1 2454 . . 3 ([y / x]x = By = B)
53, 4bitr3i 242 . 2 ([̣y / xx = By = B)
61, 2, 5vtoclbg 2916 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 3047
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 2479  df-v 2862  df-sbc 3048
This theorem is referenced by:  sbceqal  3098  eqsbc2  3104
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