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Theorem ssintab 4855
 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 4854 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦 ∈ {𝑥𝜑}𝐴𝑦)
2 sseq2 3941 . . 3 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
32ralab2 3636 . 2 (∀𝑦 ∈ {𝑥𝜑}𝐴𝑦 ↔ ∀𝑥(𝜑𝐴𝑥))
41, 3bitri 278 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  {cab 2776  ∀wral 3106   ⊆ wss 3881  ∩ cint 4838 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-in 3888  df-ss 3898  df-int 4839 This theorem is referenced by:  ssmin  4857  ssintrab  4861  intmin4  4867  dffi2  8871  rankval3b  9239  sstskm  10253  dfuzi  12061  cycsubg  18343  ssmclslem  32925  mptrcllem  40311  dfrcl2  40373  brtrclfv2  40426
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