Theorem uneq1 3283

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 2241	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	orbi1d 791	. . 3 |- ( A = B -> ( ( x e. A \/ x e. C ) <-> ( x e. B \/ x e. C ) ) )
3		elun 3277	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3277	. . 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 2175	1 |- ( A = B -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 708   = wceq 1353   e. wcel 2148   u. cun 3128

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134

This theorem is referenced by:  uneq2  3284  uneq12  3285  uneq1i  3286  uneq1d  3289  prprc1  3701  uniprg  3825  exmid1stab  4209  unexb  4443  relresfld  5159  relcoi1  5161  rdgeq2  6373  xpider  6606  findcard2  6889  findcard2s  6890  unfiexmid  6917  bdunexb  14675  bj-unexg  14676