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Theorem abssi 3418
Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssi.1  |-  ( ph  ->  x  e.  A )
Assertion
Ref Expression
abssi  |-  { x  |  ph }  C_  A
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem abssi
StepHypRef Expression
1 abssi.1 . . 3  |-  ( ph  ->  x  e.  A )
21ss2abi 3415 . 2  |-  { x  |  ph }  C_  { x  |  x  e.  A }
3 abid2 2553 . 2  |-  { x  |  x  e.  A }  =  A
42, 3sseqtri 3380 1  |-  { x  |  ph }  C_  A
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725   {cab 2422    C_ wss 3320
This theorem is referenced by:  ssab2  3427  abf  3661  intab  4080  opabss  4269  relopabi  5000  exse2  5238  opiota  6535  tfrlem8  6645  fiprc  7188  fival  7417  hartogslem1  7511  tz9.12lem1  7713  rankuni  7789  scott0  7810  r0weon  7894  alephval3  7991  aceq3lem  8001  dfac5lem4  8007  dfac2  8011  cff  8128  cfsuc  8137  cff1  8138  cflim2  8143  cfss  8145  axdc3lem  8330  axdclem  8399  gruina  8693  nqpr  8891  infcvgaux1i  12636  4sqlem1  13316  sscpwex  14015  symgval  15094  cssval  16909  hauspwpwf1  18019  itg2lcl  19619  2sqlem7  21154  nmcexi  23529  cnre2csqima  24309  colinearex  25994  itg2addnclem  26256  itg2addnc  26259  areacirc  26297  islocfin  26376  eldiophb  26815
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-in 3327  df-ss 3334
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