Theorem iinin1 4038

Description: Indexed intersection of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4021 to recover Enderton's theorem. (Contributed by Mario Carneiro, 19-Mar-2015.)

Assertion

Ref	Expression
iinin1	⊢ (A ≠ ∅ → ∩x ∈ A (C ∩ B) = (∩x ∈ A C ∩ B))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   C(x)

Proof of Theorem iinin1

Step	Hyp	Ref	Expression
1		iinin2 4037	. 2 ⊢ (A ≠ ∅ → ∩x ∈ A (B ∩ C) = (B ∩ ∩x ∈ A C))
2		incom 3449	. . . 4 ⊢ (C ∩ B) = (B ∩ C)
3	2	a1i 10	. . 3 ⊢ (x ∈ A → (C ∩ B) = (B ∩ C))
4	3	iineq2i 3989	. 2 ⊢ ∩x ∈ A (C ∩ B) = ∩x ∈ A (B ∩ C)
5		incom 3449	. 2 ⊢ (∩x ∈ A C ∩ B) = (B ∩ ∩x ∈ A C)
6	1, 4, 5	3eqtr4g 2410	1 ⊢ (A ≠ ∅ → ∩x ∈ A (C ∩ B) = (∩x ∈ A C ∩ B))

This proof depends on syntax axioms:   → wi 4   = wceq 1642   ∈ wcel 1710   ≠ wne 2517   ∩ cin 3209  ∅c0 3551  ∩ciin 3971

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 proof 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 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552  df-iin 3973

This theorem is used by: (None)