Theorem uneq1 3368

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

Assertion

Ref	Expression
uneq1	|- ( A = B -> ( A u. C ) = ( B u. C ) )

Proof of Theorem uneq1

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2298	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	orbi1d 799	. . 3 |- ( A = B -> ( ( x e. A \/ x e. C ) <-> ( x e. B \/ x e. C ) ) )
3		elun 3362	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3362	. . 3 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
5	2, 3, 4	3bitr4g 223	. 2 |- ( A = B -> ( x e. ( A u. C ) <-> x e. ( B u. C ) ) )
6	5	eqrdv 2232	1 |- ( A = B -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2205   u. cun 3211

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3217

This theorem is referenced by:  uneq2  3369  uneq12  3370  uneq1i  3371  uneq1d  3374  prprc1  3802  uniprg  3931  exmid1stab  4323  unexb  4565  relresfld  5294  relcoi1  5296  rdgeq2  6605  xpider  6842  findcard2  7148  findcard2s  7149  unfiexmid  7180  plyval  15614  bdunexb  16707  bj-unexg  16708