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Theorem abssi 3230
Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssi.1 (𝜑𝑥𝐴)
Assertion
Ref Expression
abssi {𝑥𝜑} ⊆ 𝐴
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem abssi
StepHypRef Expression
1 abssi.1 . . 3 (𝜑𝑥𝐴)
21ss2abi 3227 . 2 {𝑥𝜑} ⊆ {𝑥𝑥𝐴}
3 abid2 2298 . 2 {𝑥𝑥𝐴} = 𝐴
42, 3sseqtri 3189 1 {𝑥𝜑} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  {cab 2163  wss 3129
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-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  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  df-nfc 2308  df-in 3135  df-ss 3142
This theorem is referenced by:  ssab2  3239  abf  3466  intab  3873  opabss  4067  relopabi  4752  exse2  5002  mpoexw  6213  tfrlem8  6318  frecabex  6398  fiprc  6814  fival  6968  nqprxx  7544  ltnqex  7547  gtnqex  7548  recexprlemell  7620  recexprlemelu  7621  recexprlempr  7630  4sqlem1  12380  topnex  13517  2sqlem7  14388
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