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Theorem eluniab 3903
 Description: Membership in union of a class abstraction. (Contributed by NM, 11-Aug-1994.) (Revised by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
eluniab (A {x φ} ↔ x(A x φ))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem eluniab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 eluni 3894 . 2 (A {x φ} ↔ y(A y y {x φ}))
2 nfv 1619 . . . 4 x A y
3 nfsab1 2343 . . . 4 x y {x φ}
42, 3nfan 1824 . . 3 x(A y y {x φ})
5 nfv 1619 . . 3 y(A x φ)
6 eleq2 2414 . . . 4 (y = x → (A yA x))
7 eleq1 2413 . . . . 5 (y = x → (y {x φ} ↔ x {x φ}))
8 abid 2341 . . . . 5 (x {x φ} ↔ φ)
97, 8syl6bb 252 . . . 4 (y = x → (y {x φ} ↔ φ))
106, 9anbi12d 691 . . 3 (y = x → ((A y y {x φ}) ↔ (A x φ)))
114, 5, 10cbvex 1985 . 2 (y(A y y {x φ}) ↔ x(A x φ))
121, 11bitri 240 1 (A {x φ} ↔ x(A x φ))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  ∪cuni 3891 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-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-uni 3892 This theorem is referenced by:  elunirab  3904  dfiun2g  3999  eqtfinrelk  4486  elfv  5326  funiunfv  5467  tcfnex  6244
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