Theorem uneq1 3366

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 2296	. . . 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 3360	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3360	. . 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 2230	1 |- ( A = B -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 716   = wceq 1398   e. wcel 2203   u. cun 3209

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215

This theorem is referenced by:  uneq2  3367  uneq12  3368  uneq1i  3369  uneq1d  3372  prprc1  3800  uniprg  3929  exmid1stab  4321  unexb  4563  relresfld  5292  relcoi1  5294  rdgeq2  6603  xpider  6840  findcard2  7146  findcard2s  7147  unfiexmid  7178  plyval  15597  bdunexb  16690  bj-unexg  16691