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Theorem elintab 3937
 Description: Membership in the intersection of a class abstraction. (Contributed by NM, 30-Aug-1993.)
Hypothesis
Ref Expression
inteqab.1 A V
Assertion
Ref Expression
elintab (A {x φ} ↔ x(φA x))
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem elintab
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 inteqab.1 . . 3 A V
21elint 3932 . 2 (A {x φ} ↔ y(y {x φ} → A y))
3 nfsab1 2343 . . . 4 x y {x φ}
4 nfv 1619 . . . 4 x A y
53, 4nfim 1813 . . 3 x(y {x φ} → A y)
6 nfv 1619 . . 3 y(φA 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))
10 eleq2 2414 . . . 4 (y = x → (A yA x))
119, 10imbi12d 311 . . 3 (y = x → ((y {x φ} → A y) ↔ (φA x)))
125, 6, 11cbval 1984 . 2 (y(y {x φ} → A y) ↔ x(φA x))
132, 12bitri 240 1 (A {x φ} ↔ x(φA x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  ∩cint 3926 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-int 3927 This theorem is referenced by:  elintrab  3938  intmin4  3955  intab  3956  peano1  4402  peano2  4403  ncvspfin  4538  spfinsfincl  4539  clos1conn  5879  nchoicelem10  6298
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