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Theorem rabun2 3534
 Description: Abstraction restricted to a union. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
rabun2 {x (AB) φ} = ({x A φ} ∪ {x B φ})

Proof of Theorem rabun2
StepHypRef Expression
1 df-rab 2623 . 2 {x (AB) φ} = {x (x (AB) φ)}
2 df-rab 2623 . . . 4 {x A φ} = {x (x A φ)}
3 df-rab 2623 . . . 4 {x B φ} = {x (x B φ)}
42, 3uneq12i 3416 . . 3 ({x A φ} ∪ {x B φ}) = ({x (x A φ)} ∪ {x (x B φ)})
5 elun 3220 . . . . . . 7 (x (AB) ↔ (x A x B))
65anbi1i 676 . . . . . 6 ((x (AB) φ) ↔ ((x A x B) φ))
7 andir 838 . . . . . 6 (((x A x B) φ) ↔ ((x A φ) (x B φ)))
86, 7bitri 240 . . . . 5 ((x (AB) φ) ↔ ((x A φ) (x B φ)))
98abbii 2465 . . . 4 {x (x (AB) φ)} = {x ((x A φ) (x B φ))}
10 unab 3521 . . . 4 ({x (x A φ)} ∪ {x (x B φ)}) = {x ((x A φ) (x B φ))}
119, 10eqtr4i 2376 . . 3 {x (x (AB) φ)} = ({x (x A φ)} ∪ {x (x B φ)})
124, 11eqtr4i 2376 . 2 ({x A φ} ∪ {x B φ}) = {x (x (AB) φ)}
131, 12eqtr4i 2376 1 {x (AB) φ} = ({x A φ} ∪ {x B φ})
 Colors of variables: wff setvar class Syntax hints:   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {cab 2339  {crab 2618   ∪ cun 3207 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-rab 2623  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214 This theorem is referenced by: (None)
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