Theorem indifcom 4244

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 4170	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2	1	difeq1i 4085	. 2 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
3		indif2 4242	. 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
4		indif2 4242	. 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
5	2, 3, 4	3eqtr4i 2802	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1567   ∖ cdif 3910   ∩ cin 3912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-rab 3424  df-v 3465  df-dif 3916  df-in 3920

This theorem is referenced by:  ufprim  24035  cmmbl  25662  unmbl  25665  volinun  25674  limciun  26022  caragenuncllem  47118