MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elintab Structured version   Visualization version   GIF version

Theorem elintab 4620
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 4615 . 2 (𝐴 {𝑥𝜑} ↔ ∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦))
3 nfsab1 2760 . . . 4 𝑥 𝑦 ∈ {𝑥𝜑}
4 nfv 1994 . . . 4 𝑥 𝐴𝑦
53, 4nfim 1976 . . 3 𝑥(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦)
6 nfv 1994 . . 3 𝑦(𝜑𝐴𝑥)
7 eleq1w 2832 . . . . 5 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝑥 ∈ {𝑥𝜑}))
8 abid 2758 . . . . 5 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
97, 8syl6bb 276 . . . 4 (𝑦 = 𝑥 → (𝑦 ∈ {𝑥𝜑} ↔ 𝜑))
10 eleq2w 2833 . . . 4 (𝑦 = 𝑥 → (𝐴𝑦𝐴𝑥))
119, 10imbi12d 333 . . 3 (𝑦 = 𝑥 → ((𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ (𝜑𝐴𝑥)))
125, 6, 11cbval 2431 . 2 (∀𝑦(𝑦 ∈ {𝑥𝜑} → 𝐴𝑦) ↔ ∀𝑥(𝜑𝐴𝑥))
132, 12bitri 264 1 (𝐴 {𝑥𝜑} ↔ ∀𝑥(𝜑𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628  wcel 2144  {cab 2756  Vcvv 3349   cint 4609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-int 4610
This theorem is referenced by:  elintrab  4621  intmin4  4638  intab  4639  intid  5054  dfom3  8707  dfom5  8710  tc2  8781  dfnn2  11234  brintclab  13949  efgi  18338  efgi2  18344  mclsax  31798  heibor1lem  33933  elmapintab  38421  intabssd  38435  cotrintab  38440  dffrege76  38752
  Copyright terms: Public domain W3C validator