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Theorem elint 4955
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 2819 . . . . 5 (𝑦 = 𝐴 → (𝑦𝑥𝐴𝑥))
32imbi2d 339 . . . 4 (𝑦 = 𝐴 → ((𝑥𝐵𝑦𝑥) ↔ (𝑥𝐵𝐴𝑥)))
43albidv 1921 . . 3 (𝑦 = 𝐴 → (∀𝑥(𝑥𝐵𝑦𝑥) ↔ ∀𝑥(𝑥𝐵𝐴𝑥)))
5 df-int 4950 . . 3 𝐵 = {𝑦 ∣ ∀𝑥(𝑥𝐵𝑦𝑥)}
64, 5elab2g 3669 . 2 (𝐴 ∈ V → (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥)))
71, 6ax-mp 5 1 (𝐴 𝐵 ↔ ∀𝑥(𝑥𝐵𝐴𝑥))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537   = wceq 1539  wcel 2104  Vcvv 3472   cint 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-int 4950
This theorem is referenced by:  elint2  4956  elintabOLD  4962  intss1  4966  intun  4983  intprg  4984  intprOLD  4986  cssmre  21465  elintfv  35040  dfom5b  35188
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