Theorem difeq2 1583

Description: Equality theorem for class difference.

Assertion

Ref	Expression
difeq2	⊢ (A = B → (C ∖ A) = (C ∖ B))

Proof of Theorem difeq2

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . . 5 ⊢ (A = B → (x ∈ A ↔ x ∈ B))
2	1	negbid 463	. . . 4 ⊢ (A = B → (¬ x ∈ A ↔ ¬ x ∈ B))
3	2	anbi2d 468	. . 3 ⊢ (A = B → ((x ∈ C ∧ ¬ x ∈ A) ↔ (x ∈ C ∧ ¬ x ∈ B)))
4	3	biabdv 1183	. 2 ⊢ (A = B → {x∣(x ∈ C ∧ ¬ x ∈ A)} = {x∣(x ∈ C ∧ ¬ x ∈ B)})
5		df-dif 1489	. 2 ⊢ (C ∖ A) = {x∣(x ∈ C ∧ ¬ x ∈ A)}
6		df-dif 1489	. 2 ⊢ (C ∖ B) = {x∣(x ∈ C ∧ ¬ x ∈ B)}
7	4, 5, 6	3eqtr4g 1147	1 ⊢ (A = B → (C ∖ A) = (C ∖ B))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  {cab 1090   = wceq>/A> 1091   ∈ wcel 1092   ∖ cdif 1484

This theorem is referenced by:  difeq2i 1585  difeq2d 1588  oev 3122  sbthlem2 3350  sbth 3359  phplem5 3407  unfilem3 3440  numthlem 3598  numth 3599

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-dif 1489

metamath.org