Theorem uneq2 3091

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 3090	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uncom 3087	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3087	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3eqtr4g 2097	1 |- ( A = B -> ( C u. A ) = ( C u. B ) )

Syntax hints:   -> wi 4   = wceq 1243   u. cun 2915

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922

This theorem is referenced by:  uneq12  3092  uneq2i  3094  uneq2d  3097  uneqin  3188  disjssun  3285  uniprg  3595  sucprc  4149  unexb  4177  bdunexb  10040  bj-unexg  10041