Theorem uneq2 3219

Description: Equality theorem for the union of two classes. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
uneq2	|- ( A = B -> ( C u. A ) = ( C u. B ) )

Proof of Theorem uneq2

Step	Hyp	Ref	Expression
1		uneq1 3218	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uncom 3215	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3215	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3eqtr4g 2195	1 |- ( A = B -> ( C u. A ) = ( C u. B ) )

Syntax hints:   -> wi 4   = wceq 1331   u. cun 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070

This theorem is referenced by:  uneq12  3220  uneq2i  3222  uneq2d  3225  uneqin  3322  disjssun  3421  uniprg  3746  sucprc  4329  unexb  4358  unfiexmid  6799  unfidisj  6803  hashunlem  10543  bdunexb  13107  bj-unexg  13108