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Theorem abbi2i 2232
Description: Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
Hypothesis
Ref Expression
abbiri.1 (𝑥𝐴𝜑)
Assertion
Ref Expression
abbi2i 𝐴 = {𝑥𝜑}
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abbi2i
StepHypRef Expression
1 abeq2 2226 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 abbiri.1 . 2 (𝑥𝐴𝜑)
31, 2mpgbir 1414 1 𝐴 = {𝑥𝜑}
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1316  wcel 1465  {cab 2103
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-11 1469  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113
This theorem is referenced by:  abid2  2238  cbvralcsf  3032  cbvrexcsf  3033  cbvreucsf  3034  cbvrabcsf  3035  symdifxor  3312  dfnul2  3335  dfpr2  3516  dftp2  3542  0iin  3841  pwpwab  3870  epse  4234  fv3  5412  fo1st  6023  fo2nd  6024  xp2  6039  tfrlem3  6176  tfr1onlem3  6203  mapsn  6552  ixpconstg  6569  ixp0x  6588  nnzrab  9046  nn0zrab  9047
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