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Theorem elintrab 3618
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
Hypothesis
Ref Expression
inteqab.1 A V
Assertion
Ref Expression
elintrab (A {x Bφ} ↔ x B (φA x))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   B(x)

Proof of Theorem elintrab
StepHypRef Expression
1 inteqab.1 . . . 4 A V
21elintab 3617 . . 3 (A {x ∣ (x B φ)} ↔ x((x B φ) → A x))
3 impexp 250 . . . 4 (((x B φ) → A x) ↔ (x B → (φA x)))
43albii 1356 . . 3 (x((x B φ) → A x) ↔ x(x B → (φA x)))
52, 4bitri 173 . 2 (A {x ∣ (x B φ)} ↔ x(x B → (φA x)))
6 df-rab 2309 . . . 4 {x Bφ} = {x ∣ (x B φ)}
76inteqi 3610 . . 3 {x Bφ} = {x ∣ (x B φ)}
87eleq2i 2101 . 2 (A {x Bφ} ↔ A {x ∣ (x B φ)})
9 df-ral 2305 . 2 (x B (φA x) ↔ x(x B → (φA x)))
105, 8, 93bitr4i 201 1 (A {x Bφ} ↔ x B (φA x))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   wcel 1390  {cab 2023  wral 2300  {crab 2304  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-ral 2305  df-rab 2309  df-v 2553  df-int 3607
This theorem is referenced by:  elintrabg  3619  intmin  3626  bj-indint  9320
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