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Theorem ssint 3942
 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 Bx B A x)
Distinct variable groups:   x,A   x,B

Proof of Theorem ssint
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 dfss3 3263 . 2 (A By A y B)
2 vex 2862 . . . 4 y V
32elint2 3933 . . 3 (y Bx B y x)
43ralbii 2638 . 2 (y A y By A x B y x)
5 ralcom 2771 . . 3 (y A x B y xx B y A y x)
6 dfss3 3263 . . . 4 (A xy A y x)
76ralbii 2638 . . 3 (x B A xx B y A y x)
85, 7bitr4i 243 . 2 (y A x B y xx B A x)
91, 4, 83bitri 262 1 (A Bx B A x)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2614   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by:  ssintab  3943  ssintub  3944  iinpw  4054  fint  5245
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