Theorem indifcom 4071

Description: Commutation law for intersection and difference. (Contributed by Scott Fenton, 18-Feb-2013.)

Assertion

Ref	Expression
indifcom	⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))

Proof of Theorem indifcom

Step	Hyp	Ref	Expression
1		incom 4001	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2	1	difeq1i 3920	. 2 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
3		indif2 4069	. 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
4		indif2 4069	. 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
5	2, 3, 4	3eqtr4i 2829	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1653   ∖ cdif 3764   ∩ cin 3766

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-in 3774

This theorem is referenced by:  ufprim  22038  cmmbl  23639  unmbl  23642  volinun  23651  limciun  23996  caragenuncllem  41460