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Theorem abeq2i 2288
Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)
Hypothesis
Ref Expression
abeqi.1 𝐴 = {𝑥𝜑}
Assertion
Ref Expression
abeq2i (𝑥𝐴𝜑)

Proof of Theorem abeq2i
StepHypRef Expression
1 abeqi.1 . . 3 𝐴 = {𝑥𝜑}
21eleq2i 2244 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝜑})
3 abid 2165 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
42, 3bitri 184 1 (𝑥𝐴𝜑)
Colors of variables: wff set class
Syntax hints:  wb 105   = wceq 1353  wcel 2148  {cab 2163
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-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173
This theorem is referenced by:  rabid  2652  vex  2740  csbco  3067  csbcow  3068  csbnestgf  3109  ifmdc  3574  pwss  3591  snsspw  3764  iunpw  4480  ordon  4485  funcnv3  5278  tfrlem4  6313  tfrlem8  6318  tfrlem9  6319  tfrlemibxssdm  6327  tfr1onlembxssdm  6343  tfrcllembxssdm  6356  ixpm  6729  mapsnen  6810  sbthlem1  6955  1idprl  7588  1idpru  7589  recexprlem1ssl  7631  recexprlem1ssu  7632  recexprlemss1l  7633  recexprlemss1u  7634  txbas  13728
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