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Theorem elintrab 3792
 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 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
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 3791 . . 3 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥))
3 impexp 261 . . . 4 (((𝑥𝐵𝜑) → 𝐴𝑥) ↔ (𝑥𝐵 → (𝜑𝐴𝑥)))
43albii 1447 . . 3 (∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
52, 4bitri 183 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
6 df-rab 2426 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
76inteqi 3784 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
87eleq2i 2207 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
9 df-ral 2422 . 2 (∀𝑥𝐵 (𝜑𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
105, 8, 93bitr4i 211 1 (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1330   ∈ wcel 1481  {cab 2126  ∀wral 2417  {crab 2421  Vcvv 2690  ∩ cint 3780 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2692  df-int 3781 This theorem is referenced by:  elintrabg  3793  intmin  3800  bj-indint  13320
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