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Theorem abssi 3203
 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 3200 . 2 {𝑥𝜑} ⊆ {𝑥𝑥𝐴}
3 abid2 2278 . 2 {𝑥𝑥𝐴} = 𝐴
42, 3sseqtri 3162 1 {𝑥𝜑} ⊆ 𝐴
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 2128  {cab 2143   ⊆ wss 3102 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-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-in 3108  df-ss 3115 This theorem is referenced by:  ssab2  3212  abf  3437  intab  3836  opabss  4028  relopabi  4709  exse2  4957  tfrlem8  6259  frecabex  6339  fiprc  6753  fival  6907  nqprxx  7449  ltnqex  7452  gtnqex  7453  recexprlemell  7525  recexprlemelu  7526  recexprlempr  7535  topnex  12446
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