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Theorem csb2 3684
 Description: Alternate expression for the proper substitution into a class, without referencing substitution into a wff. Note that 𝑥 can be free in 𝐵 but cannot occur in 𝐴. (Contributed by NM, 2-Dec-2013.)
Assertion
Ref Expression
csb2 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem csb2
StepHypRef Expression
1 df-csb 3683 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbc5 3612 . . 3 ([𝐴 / 𝑥]𝑦𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑦𝐵))
32abbii 2888 . 2 {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
41, 3eqtri 2793 1 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  [wsbc 3587  ⦋csb 3682 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-v 3353  df-sbc 3588  df-csb 3683 This theorem is referenced by: (None)
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