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Theorem csbsng 3616
 Description: Distribute proper substitution through the singleton of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbsng (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})

Proof of Theorem csbsng
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbabg 3088 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦[𝐴 / 𝑥]𝑦 = 𝐵})
2 sbceq2g 3049 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵))
32abbidv 2272 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
41, 3eqtrd 2187 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦𝑦 = 𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵})
5 df-sn 3562 . . 3 {𝐵} = {𝑦𝑦 = 𝐵}
65csbeq2i 3054 . 2 𝐴 / 𝑥{𝐵} = 𝐴 / 𝑥{𝑦𝑦 = 𝐵}
7 df-sn 3562 . 2 {𝐴 / 𝑥𝐵} = {𝑦𝑦 = 𝐴 / 𝑥𝐵}
84, 6, 73eqtr4g 2212 1 (𝐴𝑉𝐴 / 𝑥{𝐵} = {𝐴 / 𝑥𝐵})
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 2125  {cab 2140  [wsbc 2933  ⦋csb 3027  {csn 3556 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-sbc 2934  df-csb 3028  df-sn 3562 This theorem is referenced by: (None)
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