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Theorem ssintab 4924
Description: Subclass of the intersection of a class abstraction. (Contributed by NM, 31-Jul-2006.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
ssintab (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ssintab
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 ssint 4923 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦 ∈ {𝑥𝜑}𝐴𝑦)
2 sseq2 3968 . . 3 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
32ralab2 3653 . 2 (∀𝑦 ∈ {𝑥𝜑}𝐴𝑦 ↔ ∀𝑥(𝜑𝐴𝑥))
41, 3bitri 274 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  {cab 2713  wral 3062  wss 3908   cint 4905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-v 3445  df-in 3915  df-ss 3925  df-int 4906
This theorem is referenced by:  ssmin  4926  ssintrab  4930  intmin4  4936  dffi2  9355  dfttrcl2  9656  rankval3b  9758  sstskm  10774  dfuzi  12590  cycsubg  18992  ssmclslem  34028  mptrcllem  41827  dfrcl2  41888  brtrclfv2  41941
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