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Theorem abbi2i 2292
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 2286 . 2 (𝐴 = {𝑥𝜑} ↔ ∀𝑥(𝑥𝐴𝜑))
2 abbiri.1 . 2 (𝑥𝐴𝜑)
31, 2mpgbir 1453 1 𝐴 = {𝑥𝜑}
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-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-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173
This theorem is referenced by:  abid2  2298  cbvralcsf  3119  cbvrexcsf  3120  cbvreucsf  3121  cbvrabcsf  3122  symdifxor  3401  dfnul2  3424  dfpr2  3611  dftp2  3641  0iin  3945  pwpwab  3974  epse  4342  fv3  5538  fo1st  6157  fo2nd  6158  xp2  6173  tfrlem3  6311  tfr1onlem3  6338  mapsn  6689  ixpconstg  6706  ixp0x  6725  nnzrab  9276  nn0zrab  9277
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