Theorem difeq2 2150

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 1532	. . . . 5 |- (A = B -> (x e. A <-> x e. B))
2	1	negbid 610	. . . 4 |- (A = B -> (-. x e. A <-> -. x e. B))
3	2	anbi2d 615	. . 3 |- (A = B -> ((x e. C /\ -. x e. A) <-> (x e. C /\ -. x e. B)))
4	3	abbidv 1574	. 2 |- (A = B -> {x | (x e. C /\ -. x e. A)} = {x | (x e. C /\ -. x e. B)})
5		df-dif 2045	. 2 |- (C \ A) = {x | (x e. C /\ -. x e. A)}
6		df-dif 2045	. 2 |- (C \ B) = {x | (x e. C /\ -. x e. B)}
7	4, 5, 6	3eqtr4g 1528	1 |- (A = B -> (C \ A) = (C \ B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  {cab 1461   \ cdif 2040

This theorem is referenced by:  difeq2i 2152  difeq2d 2155  oev 4143  sbthlem2 4434  sbth 4443  phplem4 4497  unfilem3 4532  numthlem 4763  numth 4764  fctop 7600  cctop 7602  iscld 7619  islp2 7697

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-dif 2045

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