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Theorem elintab 3617
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 {xφ} ↔ 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 3612 . 2 (A {xφ} ↔ y(y {xφ} → A y))
3 nfsab1 2027 . . . 4 x y {xφ}
4 nfv 1418 . . . 4 x A y
53, 4nfim 1461 . . 3 x(y {xφ} → A y)
6 nfv 1418 . . 3 y(φA x)
7 eleq1 2097 . . . . 5 (y = x → (y {xφ} ↔ x {xφ}))
8 abid 2025 . . . . 5 (x {xφ} ↔ φ)
97, 8syl6bb 185 . . . 4 (y = x → (y {xφ} ↔ φ))
10 eleq2 2098 . . . 4 (y = x → (A yA x))
119, 10imbi12d 223 . . 3 (y = x → ((y {xφ} → A y) ↔ (φA x)))
125, 6, 11cbval 1634 . 2 (y(y {xφ} → A y) ↔ x(φA x))
132, 12bitri 173 1 (A {xφ} ↔ x(φA x))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98  wal 1240   wcel 1390  {cab 2023  Vcvv 2551   cint 3606
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-bnd 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-v 2553  df-int 3607
This theorem is referenced by:  elintrab  3618  intmin4  3634  intab  3635  intid  3951
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