Theorem indifcom 3477

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 3421	. . 3 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
2	1	difeq1i 3343	. 2 ⊢ ((𝐴 ∩ 𝐵) ∖ 𝐶) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
3		indif2 3475	. 2 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = ((𝐴 ∩ 𝐵) ∖ 𝐶)
4		indif2 3475	. 2 ⊢ (𝐵 ∩ (𝐴 ∖ 𝐶)) = ((𝐵 ∩ 𝐴) ∖ 𝐶)
5	2, 3, 4	3eqtr4i 2269	1 ⊢ (𝐴 ∩ (𝐵 ∖ 𝐶)) = (𝐵 ∩ (𝐴 ∖ 𝐶))

Syntax hints:   = wceq 1402   ∖ cdif 3217   ∩ cin 3219

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-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-dif 3222  df-in 3226

This theorem is referenced by: (None)