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Theorem elint 4615
Description: Membership in class intersection. (Contributed by NM, 21-May-1994.)
Hypothesis
Ref Expression
elint.1 𝐴 ∈ V
Assertion
Ref Expression
elint (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem elint
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elint.1 . 2 𝐴 ∈ V
2 eleq1 2837 . . . 4 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 329 . . 3 (𝑦 = 𝐴 → ((𝑥𝐵𝑦𝑥) ↔ (𝑥𝐵𝐴𝑥)))
43albidv 2000 . 2 (𝑦 = 𝐴 → (∀𝑥(𝑥𝐵𝑦𝑥) ↔ ∀𝑥(𝑥𝐵𝐴𝑥)))
5 df-int 4610 . 2 𝐵 = {𝑦 ∣ ∀𝑥(𝑥𝐵𝑦𝑥)}
61, 4, 5elab2 3503 1 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1628   = wceq 1630  wcel 2144  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:  elint2  4616  elintab  4620  intss1  4624  intun  4641  intpr  4642  cssmre  20253  elintfv  31994  dfom5b  32350
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