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Theorem ssint 3823
Description: Subclass of a class intersection. Theorem 5.11(viii) of [Monk1] p. 52 and its converse. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
ssint (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem ssint
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfss3 3118 . 2 (𝐴 𝐵 ↔ ∀𝑦𝐴 𝑦 𝐵)
2 vex 2715 . . . 4 𝑦 ∈ V
32elint2 3814 . . 3 (𝑦 𝐵 ↔ ∀𝑥𝐵 𝑦𝑥)
43ralbii 2463 . 2 (∀𝑦𝐴 𝑦 𝐵 ↔ ∀𝑦𝐴𝑥𝐵 𝑦𝑥)
5 ralcom 2620 . . 3 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
6 dfss3 3118 . . . 4 (𝐴𝑥 ↔ ∀𝑦𝐴 𝑦𝑥)
76ralbii 2463 . . 3 (∀𝑥𝐵 𝐴𝑥 ↔ ∀𝑥𝐵𝑦𝐴 𝑦𝑥)
85, 7bitr4i 186 . 2 (∀𝑦𝐴𝑥𝐵 𝑦𝑥 ↔ ∀𝑥𝐵 𝐴𝑥)
91, 4, 83bitri 205 1 (𝐴 𝐵 ↔ ∀𝑥𝐵 𝐴𝑥)
Colors of variables: wff set class
Syntax hints:  wb 104  wcel 2128  wral 2435  wss 3102   cint 3807
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 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-in 3108  df-ss 3115  df-int 3808
This theorem is referenced by:  ssintab  3824  ssintub  3825  iinpw  3939  trint  4077  fintm  5354  bj-ssom  13498
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