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Theorem unssd 3439
 Description: A deduction showing the union of two subclasses is a subclass. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
Hypotheses
Ref Expression
unssd.1 (φA C)
unssd.2 (φB C)
Assertion
Ref Expression
unssd (φ → (AB) C)

Proof of Theorem unssd
StepHypRef Expression
1 unssd.1 . 2 (φA C)
2 unssd.2 . 2 (φB C)
3 unss 3437 . . 3 ((A C B C) ↔ (AB) C)
43biimpi 186 . 2 ((A C B C) → (AB) C)
51, 2, 4syl2anc 642 1 (φ → (AB) C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   ∪ cun 3207   ⊆ wss 3257 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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:  nchoicelem6  6294  frecsuc  6322
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