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Theorem elintrab 4955
Description: Membership in the intersection of a class abstraction. (Contributed by NM, 17-Oct-1999.)
Hypothesis
Ref Expression
elintab.ex 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 elintab.ex . . . 4 𝐴 ∈ V
21elintab 4953 . . 3 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥))
3 impexp 450 . . . 4 (((𝑥𝐵𝜑) → 𝐴𝑥) ↔ (𝑥𝐵 → (𝜑𝐴𝑥)))
43albii 1813 . . 3 (∀𝑥((𝑥𝐵𝜑) → 𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
52, 4bitri 275 . 2 (𝐴 {𝑥 ∣ (𝑥𝐵𝜑)} ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
6 df-rab 3425 . . . 4 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
76inteqi 4945 . . 3 {𝑥𝐵𝜑} = {𝑥 ∣ (𝑥𝐵𝜑)}
87eleq2i 2817 . 2 (𝐴 {𝑥𝐵𝜑} ↔ 𝐴 {𝑥 ∣ (𝑥𝐵𝜑)})
9 df-ral 3054 . 2 (∀𝑥𝐵 (𝜑𝐴𝑥) ↔ ∀𝑥(𝑥𝐵 → (𝜑𝐴𝑥)))
105, 8, 93bitr4i 303 1 (𝐴 {𝑥𝐵𝜑} ↔ ∀𝑥𝐵 (𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wal 1531  wcel 2098  {cab 2701  wral 3053  {crab 3424  Vcvv 3466   cint 4941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-int 4942
This theorem is referenced by:  elintrabg  4956  intmin  4963  naddelim  8682  rankunb  9842  isf34lem4  10369  ist1-3  23197  filufint  23768  elspani  31291  fldgensdrg  32897  evls1fldgencl  33253  ldsysgenld  33678  ldgenpisyslem1  33681  kur14lem9  34723  pclclN  39266  elpclN  39267  naddwordnexlem4  42702  lcosslsp  47368
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