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Theorem iinin1 4037
 Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4020 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)
Assertion
Ref Expression
iinin1 (Ax A (CB) = (x A CB))
Distinct variable groups:   x,A   x,B
Allowed substitution hint:   C(x)

Proof of Theorem iinin1
StepHypRef Expression
1 iinin2 4036 . 2 (Ax A (BC) = (Bx A C))
2 incom 3448 . . . 4 (CB) = (BC)
32a1i 10 . . 3 (x A → (CB) = (BC))
43iineq2i 3988 . 2 x A (CB) = x A (BC)
5 incom 3448 . 2 (x A CB) = (Bx A C)
61, 4, 53eqtr4g 2410 1 (Ax A (CB) = (x A CB))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710   ≠ wne 2516   ∩ cin 3208  ∅c0 3550  ∩ciin 3970 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551  df-iin 3972 This theorem is referenced by: (None)
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