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Theorem inss 3484
 Description: Inclusion of an intersection of two classes. (Contributed by NM, 30-Oct-2014.)
Assertion
Ref Expression
inss ((A C B C) → (AB) C)

Proof of Theorem inss
StepHypRef Expression
1 ssinss1 3483 . 2 (A C → (AB) C)
2 incom 3448 . . 3 (AB) = (BA)
3 ssinss1 3483 . . 3 (B C → (BA) C)
42, 3syl5eqss 3315 . 2 (B C → (AB) C)
51, 4jaoi 368 1 ((A C B C) → (AB) C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 357   ∩ 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: (None)
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