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Theorem elunirab 3904
 Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
elunirab (A {x B φ} ↔ x B (A x φ))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elunirab
StepHypRef Expression
1 eluniab 3903 . 2 (A {x (x B φ)} ↔ x(A x (x B φ)))
2 df-rab 2623 . . . 4 {x B φ} = {x (x B φ)}
32unieqi 3901 . . 3 {x B φ} = {x (x B φ)}
43eleq2i 2417 . 2 (A {x B φ} ↔ A {x (x B φ)})
5 df-rex 2620 . . 3 (x B (A x φ) ↔ x(x B (A x φ)))
6 an12 772 . . . 4 ((x B (A x φ)) ↔ (A x (x B φ)))
76exbii 1582 . . 3 (x(x B (A x φ)) ↔ x(A x (x B φ)))
85, 7bitri 240 . 2 (x B (A x φ) ↔ x(A x (x B φ)))
91, 4, 83bitr4i 268 1 (A {x B φ} ↔ x B (A x φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   ∈ wcel 1710  {cab 2339  ∃wrex 2615  {crab 2618  ∪cuni 3891 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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rex 2620  df-rab 2623  df-v 2861  df-uni 3892 This theorem is referenced by: (None)
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