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Theorem csb2 3061
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 3060 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 sbc5 2988 . . 3 ([𝐴 / 𝑥]𝑦𝐵 ↔ ∃𝑥(𝑥 = 𝐴𝑦𝐵))
32abbii 2293 . 2 {𝑦[𝐴 / 𝑥]𝑦𝐵} = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
41, 3eqtri 2198 1 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑥(𝑥 = 𝐴𝑦𝐵)}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wex 1492  wcel 2148  {cab 2163  [wsbc 2964  csb 3059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-sbc 2965  df-csb 3060
This theorem is referenced by: (None)
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