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Theorem rabab 2758
Description: A class abstraction restricted to the universe is unrestricted. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Assertion
Ref Expression
rabab {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}

Proof of Theorem rabab
StepHypRef Expression
1 df-rab 2464 . 2 {𝑥 ∈ V ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
2 vex 2740 . . . 4 𝑥 ∈ V
32biantrur 303 . . 3 (𝜑 ↔ (𝑥 ∈ V ∧ 𝜑))
43abbii 2293 . 2 {𝑥𝜑} = {𝑥 ∣ (𝑥 ∈ V ∧ 𝜑)}
51, 4eqtr4i 2201 1 {𝑥 ∈ V ∣ 𝜑} = {𝑥𝜑}
Colors of variables: wff set class
Syntax hints:  wa 104   = wceq 1353  wcel 2148  {cab 2163  {crab 2459  Vcvv 2737
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-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  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-rab 2464  df-v 2739
This theorem is referenced by:  notab  3405  intmin2  3868  euen1  6796  bj-omind  14335
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