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Theorem elintrab 3938
 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 B φ} ↔ 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 3937 . . 3 (A {x (x B φ)} ↔ x((x B φ) → A x))
3 impexp 433 . . . 4 (((x B φ) → A x) ↔ (x B → (φA x)))
43albii 1566 . . 3 (x((x B φ) → A x) ↔ x(x B → (φA x)))
52, 4bitri 240 . 2 (A {x (x B φ)} ↔ x(x B → (φA x)))
6 df-rab 2623 . . . 4 {x B φ} = {x (x B φ)}
76inteqi 3930 . . 3 {x B φ} = {x (x B φ)}
87eleq2i 2417 . 2 (A {x B φ} ↔ A {x (x B φ)})
9 df-ral 2619 . 2 (x B (φA x) ↔ x(x B → (φA x)))
105, 8, 93bitr4i 268 1 (A {x B φ} ↔ x B (φA x))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710  {cab 2339  ∀wral 2614  {crab 2618  Vcvv 2859  ∩cint 3926 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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-ral 2619  df-rab 2623  df-v 2861  df-int 3927 This theorem is referenced by:  elintrabg  3939  intmin  3946
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