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Theorem abssi 3341
 Description: Inference of abstraction subclass from implication. (Contributed by NM, 20-Jan-2006.)
Hypothesis
Ref Expression
abssi.1 (φx A)
Assertion
Ref Expression
abssi {x φ} A
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem abssi
StepHypRef Expression
1 abssi.1 . . 3 (φx A)
21ss2abi 3338 . 2 {x φ} {x x A}
3 abid2 2470 . 2 {x x A} = A
42, 3sseqtri 3303 1 {x φ} A
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1710  {cab 2339   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  ssab2  3350  abf  3584  intab  3956  opkabssvvk  4208  fvclss  5462  mapsspw  6022  spacssnc  6284
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