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Theorem ss2rabdv 4076
Description: Deduction of restricted abstraction subclass from implication. (Contributed by NM, 30-May-2006.)
Hypothesis
Ref Expression
ss2rabdv.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
ss2rabdv (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem ss2rabdv
StepHypRef Expression
1 ss2rabdv.1 . . 3 ((𝜑𝑥𝐴) → (𝜓𝜒))
21ralrimiva 3146 . 2 (𝜑 → ∀𝑥𝐴 (𝜓𝜒))
3 ss2rab 4071 . 2 ({𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒} ↔ ∀𝑥𝐴 (𝜓𝜒))
42, 3sylibr 234 1 (𝜑 → {𝑥𝐴𝜓} ⊆ {𝑥𝐴𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2108  wral 3061  {crab 3436  wss 3951
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rab 3437  df-ss 3968
This theorem is referenced by:  rabssrabd  4083  sess1  5650  suppssov1  8222  suppssov2  8223  suppssfv  8227  cofon1  8710  naddssim  8723  harword  9603  scottex  9925  mrcss  17659  mndpsuppss  18778  ablfac1b  20090  mptscmfsupp0  20925  lspss  20982  dsmmacl  21761  dsmmsubg  21763  dsmmlss  21764  aspss  21897  psdmul  22170  scmatdmat  22521  clsss  23062  lfinpfin  23532  qustgpopn  24128  metss2lem  24524  equivcau  25334  rrxmvallem  25438  ovolsslem  25519  itg2monolem1  25785  lgamucov  27081  sqff1o  27225  musum  27234  madess  27915  cofcut1  27954  cusgrfilem1  29473  clwlknf1oclwwlknlem3  30102  occon  31306  spanss  31367  rmfsupp2  33242  fldgenss  33318  locfinreflem  33839  omsmon  34300  orvclteinc  34478  fin2solem  37613  poimirlem26  37653  poimirlem27  37654  cnambfre  37675  pclssN  39896  2polssN  39917  dihglblem3N  41297  dochss  41367  mapdordlem2  41639  nna4b4nsq  42670  itgoss  43175  nzss  44336  ovnsslelem  46575  gpgusgralem  48011  rmsuppss  48286  scmsuppss  48287
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