Theorem iinin1m 4040

Description: Indexed intersection of intersection. Compare to Theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Jim Kingdon, 17-Aug-2018.)

Assertion

Ref	Expression
iinin1m	⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem iinin1m

Step	Hyp	Ref	Expression
1		iinin2m 4039	. 2 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶))
2		incom 3399	. . . 4 ⊢ (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶)
3	2	a1i 9	. . 3 ⊢ (𝑥 ∈ 𝐴 → (𝐶 ∩ 𝐵) = (𝐵 ∩ 𝐶))
4	3	iineq2i 3989	. 2 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = ∩ 𝑥 ∈ 𝐴 (𝐵 ∩ 𝐶)
5		incom 3399	. 2 ⊢ (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵) = (𝐵 ∩ ∩ 𝑥 ∈ 𝐴 𝐶)
6	1, 4, 5	3eqtr4g 2289	1 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∩ 𝑥 ∈ 𝐴 (𝐶 ∩ 𝐵) = (∩ 𝑥 ∈ 𝐴 𝐶 ∩ 𝐵))

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2202   ∩ cin 3199  ∩ ciin 3971

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-in 3206  df-iin 3973

This theorem is referenced by: (None)