Theorem uneq2 3412

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 3411	. 2 |- (A = B -> (A u. C) = (B u. C))
2		uncom 3408	. 2 |- (C u. A) = (A u. C)
3		uncom 3408	. 2 |- (C u. B) = (B u. C)
4	1, 2, 3	3eqtr4g 2410	1 |- (A = B -> (C u. A) = (C u. B))

Syntax hints:   -> wi 4   = wceq 1642   u. cun 3207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-un 3214

This theorem is referenced by:  uneq12  3413  uneq2i  3415  uneq2d  3418  uneqin  3506  disjssun  3608  uniprg  3906  eladdci  4399  addcid1  4405  elsuc  4413  addcass  4415  phialllem2  4617  cupvalg  5812