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Theorem abssi 3043
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 3040 . 2 {𝑥𝜑} ⊆ {𝑥𝑥𝐴}
3 abid2 2174 . 2 {𝑥𝑥𝐴} = 𝐴
42, 3sseqtri 3005 1 {𝑥𝜑} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1409  {cab 2042  wss 2945
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-in 2952  df-ss 2959
This theorem is referenced by:  ssab2  3052  abf  3288  intab  3672  opabss  3849  relopabi  4491  exse2  4727  tfrlem8  5965  frecabex  6015  fiprc  6323  nqprxx  6702  ltnqex  6705  gtnqex  6706  recexprlemell  6778  recexprlemelu  6779  recexprlempr  6788
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