Theorem indifcom 3501

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 3449	. . 3 ⊢ (A ∩ B) = (B ∩ A)
2	1	difeq1i 3382	. 2 ⊢ ((A ∩ B) ∖ C) = ((B ∩ A) ∖ C)
3		indif2 3499	. 2 ⊢ (A ∩ (B ∖ C)) = ((A ∩ B) ∖ C)
4		indif2 3499	. 2 ⊢ (B ∩ (A ∖ C)) = ((B ∩ A) ∖ C)
5	2, 3, 4	3eqtr4i 2383	1 ⊢ (A ∩ (B ∖ C)) = (B ∩ (A ∖ C))

Syntax hints:   = wceq 1642   ∖ cdif 3207   ∩ cin 3209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216

This theorem is referenced by: (None)