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Theorem eluniab 3566
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 {xφ} ↔ 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 3557 . 2 (A {xφ} ↔ y(A y y {xφ}))
2 nfv 1402 . . . 4 x A y
3 nfsab1 2012 . . . 4 x y {xφ}
42, 3nfan 1439 . . 3 x(A y y {xφ})
5 nfv 1402 . . 3 y(A x φ)
6 eleq2 2083 . . . 4 (y = x → (A yA x))
7 eleq1 2082 . . . . 5 (y = x → (y {xφ} ↔ x {xφ}))
8 abid 2010 . . . . 5 (x {xφ} ↔ φ)
97, 8syl6bb 185 . . . 4 (y = x → (y {xφ} ↔ φ))
106, 9anbi12d 445 . . 3 (y = x → ((A y y {xφ}) ↔ (A x φ)))
114, 5, 10cbvex 1621 . 2 (y(A y y {xφ}) ↔ x(A x φ))
121, 11bitri 173 1 (A {xφ} ↔ x(A x φ))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98  wex 1362   wcel 1374  {cab 2008   cuni 3554
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-uni 3555
This theorem is referenced by:  elunirab  3567  dfiun2g  3663  inuni  3883  snnex  4131  elfv  5101  unielxp  5723  tfrlem9  5857
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