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Theorem unssbd 3441
 Description: If (A ∪ B) is contained in C, so is B. One-way deduction form of unss 3437. Partial converse of unssd 3439. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
unssad.1 (φ → (AB) C)
Assertion
Ref Expression
unssbd (φB C)

Proof of Theorem unssbd
StepHypRef Expression
1 unssad.1 . . 3 (φ → (AB) C)
2 unss 3437 . . 3 ((A C B C) ↔ (AB) C)
31, 2sylibr 203 . 2 (φ → (A C B C))
43simprd 449 1 (φB C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∪ cun 3207   ⊆ 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-un 3214  df-ss 3259 This theorem is referenced by: (None)
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