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Theorem abssi 3140
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 3137 . 2 {𝑥𝜑} ⊆ {𝑥𝑥𝐴}
3 abid2 2236 . 2 {𝑥𝑥𝐴} = 𝐴
42, 3sseqtri 3099 1 {𝑥𝜑} ⊆ 𝐴
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  {cab 2101  wss 3039
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-in 3045  df-ss 3052
This theorem is referenced by:  ssab2  3149  abf  3374  intab  3768  opabss  3960  relopabi  4633  exse2  4881  tfrlem8  6181  frecabex  6261  fiprc  6675  fival  6824  nqprxx  7318  ltnqex  7321  gtnqex  7322  recexprlemell  7394  recexprlemelu  7395  recexprlempr  7404  topnex  12161
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