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Theorem abeq2i 2164
Description: Equality of a class variable and a class abstraction (inference rule). (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 2120 . 2 (𝑥𝐴𝑥 ∈ {𝑥𝜑})
3 abid 2044 . 2 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
42, 3bitri 177 1 (𝑥𝐴𝜑)
Colors of variables: wff set class
Syntax hints:  wb 102   = wceq 1259  wcel 1409  {cab 2042
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052
This theorem is referenced by:  rabid  2502  vex  2577  csbco  2888  csbnestgf  2925  pwss  3401  snsspw  3562  iunpw  4238  ordon  4239  funcnv3  4988  tfrlem4  5959  tfrlem8  5964  tfrlem9  5965  tfrlemibxssdm  5971  1idprl  6745  1idpru  6746  recexprlem1ssl  6788  recexprlem1ssu  6789  recexprlemss1l  6790  recexprlemss1u  6791
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