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Theorem unssbd 3115
 Description: If (A ∪ B) is contained in 𝐶, so is B. One-way deduction form of unss 3111. Partial converse of unssd 3113. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
unssad.1 (φ → (AB) ⊆ 𝐶)
Assertion
Ref Expression
unssbd (φB𝐶)

Proof of Theorem unssbd
StepHypRef Expression
1 unssad.1 . . 3 (φ → (AB) ⊆ 𝐶)
2 unss 3111 . . 3 ((A𝐶 B𝐶) ↔ (AB) ⊆ 𝐶)
31, 2sylibr 137 . 2 (φ → (A𝐶 B𝐶))
43simprd 107 1 (φB𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ∪ cun 2909   ⊆ wss 2911 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925 This theorem is referenced by:  eldifpw  4174  ertr  6057
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