Theorem uneq12i 1609

Description: Equality inference for union of two classes.

Hypotheses

Ref	Expression
uneq1i.1	⊢ A = B
uneq12i.2	⊢ C = D

Assertion

Ref	Expression
uneq12i	⊢ (A ∪ C) = (B ∪ D)

Proof of Theorem uneq12i

Step	Hyp	Ref	Expression
1		uneq1i.1	. . 3 ⊢ A = B
2	1	uneq1i 1607	. 2 ⊢ (A ∪ C) = (B ∪ C)
3		uneq12i.2	. . 3 ⊢ C = D
4	3	uneq2i 1608	. 2 ⊢ (B ∪ C) = (B ∪ D)
5	2, 4	eqtr 1119	1 ⊢ (A ∪ C) = (B ∪ D)

Syntax hints:   = wceq 1091   ∪ cun 1485

This theorem is referenced by:  indir 1678  difundir 1682  difindi 1683  symdif1 1689  unrab 1694  unopab 2121  xpundi 2461  xpundir 2462  xpun 2463  resundi 2582  resundir 2583  rnun 2644  imaun 2647  df2o2 3112  sbthlem5 3353  rankpr 3536  fac0 4871  ruclem6 4890

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

metamath.org