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Theorem ssintab 4459
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 4458 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦 ∈ {𝑥𝜑}𝐴𝑦)
2 sseq2 3606 . . 3 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
32ralab2 3353 . 2 (∀𝑦 ∈ {𝑥𝜑}𝐴𝑦 ↔ ∀𝑥(𝜑𝐴𝑥))
41, 3bitri 264 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1478  {cab 2607  wral 2907  wss 3555   cint 4440
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-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-int 4441
This theorem is referenced by:  ssmin  4461  ssintrab  4465  intmin4  4471  dffi2  8273  rankval3b  8633  sstskm  9608  dfuzi  11412  cycsubg  17543  ssmclslem  31167  mptrcllem  37398  dfrcl2  37444  brtrclfv2  37497
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