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Theorem ssintab 3788
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 3787 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦 ∈ {𝑥𝜑}𝐴𝑦)
2 sseq2 3121 . . 3 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
32ralab2 2848 . 2 (∀𝑦 ∈ {𝑥𝜑}𝐴𝑦 ↔ ∀𝑥(𝜑𝐴𝑥))
41, 3bitri 183 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1329  {cab 2125  wral 2416  wss 3071   cint 3771
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-in 3077  df-ss 3084  df-int 3772
This theorem is referenced by:  ssmin  3790  ssintrab  3794  intmin4  3799  dfuzi  9173
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