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Theorem ssini 3478
 Description: An inference showing that a subclass of two classes is a subclass of their intersection. (Contributed by NM, 24-Nov-2003.)
Hypotheses
Ref Expression
ssini.1 A B
ssini.2 A C
Assertion
Ref Expression
ssini A (BC)

Proof of Theorem ssini
StepHypRef Expression
1 ssini.1 . . 3 A B
2 ssini.2 . . 3 A C
31, 2pm3.2i 441 . 2 (A B A C)
4 ssin 3477 . 2 ((A B A C) ↔ A (BC))
53, 4mpbi 199 1 A (BC)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   ∩ 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:  inv1  3577  fvfullfunlem3  5863
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