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Theorem ss2abdv 3654
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 1852 . 2 (𝜑 → ∀𝑥(𝜓𝜒))
3 ss2ab 3649 . 2 ({𝑥𝜓} ⊆ {𝑥𝜒} ↔ ∀𝑥(𝜓𝜒))
42, 3sylibr 224 1 (𝜑 → {𝑥𝜓} ⊆ {𝑥𝜒})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478  {cab 2607  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-in 3562  df-ss 3569
This theorem is referenced by:  intss  4463  ssopab2  4961  ssoprab2  6664  suppimacnvss  7250  suppimacnv  7251  ressuppss  7259  ss2ixp  7865  fiss  8274  tcss  8564  tcel  8565  infmap2  8984  cfub  9015  cflm  9016  cflecard  9019  clsslem  13657  cncmet  23027  plyss  23859  ofrn2  29281  sigaclci  29973  subfacp1lem6  30872  ss2mcls  31170  itg2addnclem  33090  sdclem1  33168  istotbnd3  33199  sstotbnd  33203  aomclem4  37104  hbtlem4  37174  hbtlem3  37175  rngunsnply  37221  iocinico  37275
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