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Theorem ss2abdv 3804
Description: Deduction of abstraction subclass from implication. (Contributed by NM, 29-Jul-2011.)
Hypothesis
Ref Expression
ss2abdv.1 (𝜑 → (𝜓𝜒))
Assertion
Ref Expression
ss2abdv (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem ss2abdv
StepHypRef Expression
1 ss2abdv.1 . . 3 (𝜑 → (𝜓𝜒))
21alrimiv 1992 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 ss2ab 3799 . 2 ({𝑥𝜓} ⊆ {𝑥𝜒} ↔ ∀𝑥(𝜓𝜒))
42, 3sylibr 224 1 (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1618  {cab 2734  wss 3703
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-in 3710  df-ss 3717
This theorem is referenced by:  intss  4638  ssopab2  5139  ssoprab2  6864  suppimacnvss  7461  suppimacnv  7462  ressuppss  7470  ss2ixp  8075  fiss  8483  tcss  8781  tcel  8782  infmap2  9203  cfub  9234  cflm  9235  cflecard  9238  clsslem  13895  cncmet  23290  plyss  24125  ofrn2  29722  sigaclci  30475  subfacp1lem6  31445  ss2mcls  31743  itg2addnclem  33743  sdclem1  33821  istotbnd3  33852  sstotbnd  33856  qsss1  34346  aomclem4  38098  hbtlem4  38167  hbtlem3  38168  rngunsnply  38214  iocinico  38268
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