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Theorem csb2 3891
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 3890 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbc5 3801 . . 3 ([𝐴 / 𝑥]𝑦𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑦𝐵))
32abbii 2795 . 2 {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
41, 3eqtri 2753 1 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wex 1773  wcel 2098  {cab 2702  [wsbc 3773  csb 3889
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-sbc 3774  df-csb 3890
This theorem is referenced by: (None)
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