ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elintab GIF version

Theorem elintab 3750
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(𝜑𝐴𝑥))
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 3745 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦))
3 nfsab1 2105 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1491 . . . 4 𝑥 𝐴𝑦
53, 4nfim 1534 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦)
6 nfv 1491 . . 3 𝑦(𝜑𝐴𝑥)
7 eleq1 2178 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2103 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 195 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
10 eleq2 2179 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
119, 10imbi12d 233 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ (𝜑𝐴𝑥)))
125, 6, 11cbval 1710 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ ∀𝑥(𝜑𝐴𝑥))
132, 12bitri 183 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1312  wcel 1463  {cab 2101  Vcvv 2658   cint 3739
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-int 3740
This theorem is referenced by:  elintrab  3751  intmin4  3767  intab  3768  intid  4114
  Copyright terms: Public domain W3C validator