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Theorem abbi2i 2254
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 2248 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 abbiri.1 . 2 (𝑥𝐴𝜑)
31, 2mpgbir 1429 1 𝐴 = {𝑥𝜑}
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1331  wcel 1480  {cab 2125
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-11 1484  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135
This theorem is referenced by:  abid2  2260  cbvralcsf  3062  cbvrexcsf  3063  cbvreucsf  3064  cbvrabcsf  3065  symdifxor  3342  dfnul2  3365  dfpr2  3546  dftp2  3572  0iin  3871  pwpwab  3900  epse  4264  fv3  5444  fo1st  6055  fo2nd  6056  xp2  6071  tfrlem3  6208  tfr1onlem3  6235  mapsn  6584  ixpconstg  6601  ixp0x  6620  nnzrab  9090  nn0zrab  9091
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