Theorem difeqri 3388

Description: Inference from membership to difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypothesis

Ref	Expression
difeqri.1	⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ x ∈ C)

Assertion

Ref	Expression
difeqri	⊢ (A ∖ B) = C

Distinct variable groups:   x,A   x,B   x,C

Proof of Theorem difeqri

Step	Hyp	Ref	Expression
1		eldif 3222	. . 3 ⊢ (x ∈ (A ∖ B) ↔ (x ∈ A ∧ ¬ x ∈ B))
2		difeqri.1	. . 3 ⊢ ((x ∈ A ∧ ¬ x ∈ B) ↔ x ∈ C)
3	1, 2	bitri 240	. 2 ⊢ (x ∈ (A ∖ B) ↔ x ∈ C)
4	3	eqriv 2350	1 ⊢ (A ∖ B) = C

Syntax hints:  ¬ wn 3   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∖ cdif 3207

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:  difdif  3393  ddif  3399  dfss4  3490  difin  3493  difab  3524