Theorem uneq1 1605

Description: Equality theorem for union of two classes.

Assertion

Ref	Expression
uneq1	|- (A = B -> (A u. C) = (B u. C))

Proof of Theorem uneq1

Step	Hyp	Ref	Expression
1		eleq2 1150	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	orbi1d 467	. . 3 |- (A = B -> ((x e. A \/ x e. C) <-> (x e. B \/ x e. C)))
3		elun 1601	. . 3 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
4		elun 1601	. . 3 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	2, 3, 4	3bitr4g 428	. 2 |- (A = B -> (x e. (A u. C) <-> x e. (B u. C)))
6	5	cleqrd 1100	1 |- (A = B -> (A u. C) = (B u. C))

Syntax hints:   -> wi 2   \/ wo 195   = wceq 1091   e. wcel 1092   u. cun 1485

This theorem is referenced by:  uneq2 1606  uneq1i 1607  uneq1d 1610  uneq12 1613  unineq 1680  prprc 1839  unexb 1950  suceq 2288  unxpdom 3650  sshjvalt 5321

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-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490

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