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Theorem ssind 3359
Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
ssind.1 (𝜑𝐴𝐵)
ssind.2 (𝜑𝐴𝐶)
Assertion
Ref Expression
ssind (𝜑𝐴 ⊆ (𝐵𝐶))

Proof of Theorem ssind
StepHypRef Expression
1 ssind.1 . 2 (𝜑𝐴𝐵)
2 ssind.2 . 2 (𝜑𝐴𝐶)
3 ssin 3357 . . 3 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
43biimpi 120 . 2 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
51, 2, 4syl2anc 411 1 (𝜑𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  cin 3128  wss 3129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-in 3135  df-ss 3142
This theorem is referenced by:  ntrin  13257  lmss  13379
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