Theorem difeq1 2153

Description: Equality theorem for class difference.

Assertion

Ref	Expression
difeq1	|- (A = B -> (A \ C) = (B \ C))

Proof of Theorem difeq1

Step	Hyp	Ref	Expression
1		eleq2 1535	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	anbi1d 617	. . 3 |- (A = B -> ((x e. A /\ -. x e. C) <-> (x e. B /\ -. x e. C)))
3	2	abbidv 1577	. 2 |- (A = B -> {x | (x e. A /\ -. x e. C)} = {x | (x e. B /\ -. x e. C)})
4		df-dif 2049	. 2 |- (A \ C) = {x | (x e. A /\ -. x e. C)}
5		df-dif 2049	. 2 |- (B \ C) = {x | (x e. B /\ -. x e. C)}
6	3, 4, 5	3eqtr4g 1531	1 |- (A = B -> (A \ C) = (B \ C))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  {cab 1463   \ cdif 2044

This theorem is referenced by:  difeq1i 2155  difeq1d 2158  kmlem9 4773  kmlem11 4775  kmlem12 4776  lpval 7743  rcfpfillem6 10595  rcfpfillem6OLD 10596

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-dif 2049

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