Theorem difeq2 2144

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 1527	. . . . 5 |- (A = B -> (x e. A <-> x e. B))
2	1	negbid 609	. . . 4 |- (A = B -> (-. x e. A <-> -. x e. B))
3	2	anbi2d 614	. . 3 |- (A = B -> ((x e. C /\ -. x e. A) <-> (x e. C /\ -. x e. B)))
4	3	abbidv 1569	. 2 |- (A = B -> {x | (x e. C /\ -. x e. A)} = {x | (x e. C /\ -. x e. B)})
5		df-dif 2039	. 2 |- (C \ A) = {x | (x e. C /\ -. x e. A)}
6		df-dif 2039	. 2 |- (C \ B) = {x | (x e. C /\ -. x e. B)}
7	4, 5, 6	3eqtr4g 1523	1 |- (A = B -> (C \ A) = (C \ B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 953   e. wcel 955  {cab 1456   \ cdif 2034

This theorem is referenced by:  difeq2i 2146  difeq2d 2149  oev 4137  sbthlem2 4428  sbth 4437  phplem4 4491  unfilem3 4526  numthlem 4755  numth 4756  fctop 7592  cctop 7594  iscld 7611  islp2 7688

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-dif 2039

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