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Theorem abssi 3231
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 3228 . 2 {𝑥𝜑} ⊆ {𝑥𝑥𝐴}
3 abid2 2298 . 2 {𝑥𝑥𝐴} = 𝐴
42, 3sseqtri 3190 1 {𝑥𝜑} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  {cab 2163  wss 3130
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 3136  df-ss 3143
This theorem is referenced by:  ssab2  3240  abf  3467  intab  3874  opabss  4068  relopabi  4753  exse2  5003  mpoexw  6214  tfrlem8  6319  frecabex  6399  fiprc  6815  fival  6969  nqprxx  7545  ltnqex  7548  gtnqex  7549  recexprlemell  7621  recexprlemelu  7622  recexprlempr  7631  4sqlem1  12386  topnex  13589  2sqlem7  14471
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