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Theorem intss 3709
Description: Intersection of subclasses. (Contributed by NM, 14-Oct-1999.)
Assertion
Ref Expression
intss (𝐴𝐵 𝐵 𝐴)

Proof of Theorem intss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 imim1 75 . . . . 5 ((𝑦𝐴𝑦𝐵) → ((𝑦𝐵𝑥𝑦) → (𝑦𝐴𝑥𝑦)))
21al2imi 1392 . . . 4 (∀𝑦(𝑦𝐴𝑦𝐵) → (∀𝑦(𝑦𝐵𝑥𝑦) → ∀𝑦(𝑦𝐴𝑥𝑦)))
3 vex 2622 . . . . 5 𝑥 ∈ V
43elint 3694 . . . 4 (𝑥 𝐵 ↔ ∀𝑦(𝑦𝐵𝑥𝑦))
53elint 3694 . . . 4 (𝑥 𝐴 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
62, 4, 53imtr4g 203 . . 3 (∀𝑦(𝑦𝐴𝑦𝐵) → (𝑥 𝐵𝑥 𝐴))
76alrimiv 1802 . 2 (∀𝑦(𝑦𝐴𝑦𝐵) → ∀𝑥(𝑥 𝐵𝑥 𝐴))
8 dfss2 3014 . 2 (𝐴𝐵 ↔ ∀𝑦(𝑦𝐴𝑦𝐵))
9 dfss2 3014 . 2 ( 𝐵 𝐴 ↔ ∀𝑥(𝑥 𝐵𝑥 𝐴))
107, 8, 93imtr4i 199 1 (𝐴𝐵 𝐵 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1287  wcel 1438  wss 2999   cint 3688
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3005  df-ss 3012  df-int 3689
This theorem is referenced by: (None)
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