Theorem uneq1 3254

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 2221	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	orbi1d 781	. . 3 |- ( A = B -> ( ( x e. A \/ x e. C ) <-> ( x e. B \/ x e. C ) ) )
3		elun 3248	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3248	. . 3 |- ( x e. ( B u. C ) <-> ( x e. B \/ x e. C ) )
5	2, 3, 4	3bitr4g 222	. 2 |- ( A = B -> ( x e. ( A u. C ) <-> x e. ( B u. C ) ) )
6	5	eqrdv 2155	1 |- ( A = B -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 698   = wceq 1335   e. wcel 2128   u. cun 3100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106

This theorem is referenced by:  uneq2  3255  uneq12  3256  uneq1i  3257  uneq1d  3260  prprc1  3667  uniprg  3787  unexb  4401  relresfld  5114  relcoi1  5116  rdgeq2  6316  xpider  6548  findcard2  6831  findcard2s  6832  unfiexmid  6859  bdunexb  13466  bj-unexg  13467  exmid1stab  13543