Theorem indifcom 4229

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 4155	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2	1	difeq1i 4070	. 2 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
3		indif2 4227	. 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
4		indif2 4227	. 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
5	2, 3, 4	3eqtr4i 2793	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3896   ∩ cin 3898

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-in 3906

This theorem is used by:  ufprim  24138  cmmbl  25765  unmbl  25768  volinun  25777  limciun  26124  caragenuncllem  47343