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Theorem elunirab 3584
 Description: Membership in union of a class abstraction. (Contributed by NM, 4-Oct-2006.)
Assertion
Ref Expression
elunirab (A {x Bφ} ↔ 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 3583 . 2 (A {x ∣ (x B φ)} ↔ x(A x (x B φ)))
2 df-rab 2309 . . . 4 {x Bφ} = {x ∣ (x B φ)}
32unieqi 3581 . . 3 {x Bφ} = {x ∣ (x B φ)}
43eleq2i 2101 . 2 (A {x Bφ} ↔ A {x ∣ (x B φ)})
5 df-rex 2306 . . 3 (x B (A x φ) ↔ x(x B (A x φ)))
6 an12 495 . . . 4 ((x B (A x φ)) ↔ (A x (x B φ)))
76exbii 1493 . . 3 (x(x B (A x φ)) ↔ x(A x (x B φ)))
85, 7bitri 173 . 2 (x B (A x φ) ↔ x(A x (x B φ)))
91, 4, 83bitr4i 201 1 (A {x Bφ} ↔ x B (A x φ))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  {cab 2023  ∃wrex 2301  {crab 2304  ∪ cuni 3571 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-rex 2306  df-rab 2309  df-v 2553  df-uni 3572 This theorem is referenced by: (None)
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