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Theorem ssind 3161
 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 3159 . . 3 ((𝐴𝐵𝐴𝐶) ↔ 𝐴 ⊆ (𝐵𝐶))
43biimpi 113 . 2 ((𝐴𝐵𝐴𝐶) → 𝐴 ⊆ (𝐵𝐶))
51, 2, 4syl2anc 391 1 (𝜑𝐴 ⊆ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∩ cin 2916   ⊆ wss 2917 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931 This theorem is referenced by: (None)
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