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Theorem ssind 3300
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 3298 . . 3 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
43biimpi 119 . 2 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
51, 2, 4syl2anc 408 1 (𝜑𝐴 ⊆ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  cin 3070  wss 3071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084
This theorem is referenced by:  ntrin  12319  lmss  12441
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