Theorem uneq1 3411

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 2414	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	orbi1d 683	. . 3 |- (A = B -> ((x e. A \/ x e. C) <-> (x e. B \/ x e. C)))
3		elun 3220	. . 3 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
4		elun 3220	. . 3 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	2, 3, 4	3bitr4g 279	. 2 |- (A = B -> (x e. (A u. C) <-> x e. (B u. C)))
6	5	eqrdv 2351	1 |- (A = B -> (A u. C) = (B u. C))

Syntax hints:   -> wi 4   \/ wo 357   = wceq 1642   e. wcel 1710   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:  uneq2  3412  uneq12  3413  uneq1i  3414  uneq1d  3417  unineq  3505  adj11  3889  uniprg  3906  pwadjoin  4119  eladdci  4399  elsuci  4414  addcass  4415  nnsucelr  4428  nnadjoin  4520  phi011  4599  cupvalg  5812  el2c  6191  nmembers1lem3  6270