Theorem uneq2 3295

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 3294	. 2 |- ( A = B -> ( A u. C ) = ( B u. C ) )
2		uncom 3291	. 2 |- ( C u. A ) = ( A u. C )
3		uncom 3291	. 2 |- ( C u. B ) = ( B u. C )
4	1, 2, 3	3eqtr4g 2245	1 |- ( A = B -> ( C u. A ) = ( C u. B ) )

Syntax hints:   -> wi 4   = wceq 1363   u. cun 3139

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-v 2751  df-un 3145

This theorem is referenced by:  uneq12  3296  uneq2i  3298  uneq2d  3301  uneqin  3398  disjssun  3498  uniprg  3836  sucprc  4424  unexb  4454  unfiexmid  6931  unfidisj  6935  hashunlem  10798  bdunexb  14968  bj-unexg  14969