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Theorem ssin 3477
 Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
Assertion
Ref Expression
ssin ((A B A C) ↔ A (BC))

Proof of Theorem ssin
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elin 3219 . . . . 5 (x (BC) ↔ (x B x C))
21imbi2i 303 . . . 4 ((x Ax (BC)) ↔ (x A → (x B x C)))
32albii 1566 . . 3 (x(x Ax (BC)) ↔ x(x A → (x B x C)))
4 jcab 833 . . . 4 ((x A → (x B x C)) ↔ ((x Ax B) (x Ax C)))
54albii 1566 . . 3 (x(x A → (x B x C)) ↔ x((x Ax B) (x Ax C)))
6 19.26 1593 . . 3 (x((x Ax B) (x Ax C)) ↔ (x(x Ax B) x(x Ax C)))
73, 5, 63bitrri 263 . 2 ((x(x Ax B) x(x Ax C)) ↔ x(x Ax (BC)))
8 dfss2 3262 . . 3 (A Bx(x Ax B))
9 dfss2 3262 . . 3 (A Cx(x Ax C))
108, 9anbi12i 678 . 2 ((A B A C) ↔ (x(x Ax B) x(x Ax C)))
11 dfss2 3262 . 2 (A (BC) ↔ x(x Ax (BC)))
127, 10, 113bitr4i 268 1 ((A B A C) ↔ A (BC))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∩ cin 3208   ⊆ wss 3257 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259 This theorem is referenced by:  ssini  3478  ssind  3479  uneqin  3506  disjpss  3601  fin  5246  clos1induct  5880  sbthlem1  6203
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