Theorem uneq1 3320

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 2269	. . . 4 |- ( A = B -> ( x e. A <-> x e. B ) )
2	1	orbi1d 793	. . 3 |- ( A = B -> ( ( x e. A \/ x e. C ) <-> ( x e. B \/ x e. C ) ) )
3		elun 3314	. . 3 |- ( x e. ( A u. C ) <-> ( x e. A \/ x e. C ) )
4		elun 3314	. . 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 2203	1 |- ( A = B -> ( A u. C ) = ( B u. C ) )

Syntax hints:   -> wi 4   \/ wo 710   = wceq 1373   e. wcel 2176   u. cun 3164

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170

This theorem is referenced by:  uneq2  3321  uneq12  3322  uneq1i  3323  uneq1d  3326  prprc1  3741  uniprg  3865  exmid1stab  4252  unexb  4489  relresfld  5212  relcoi1  5214  rdgeq2  6458  xpider  6693  findcard2  6986  findcard2s  6987  unfiexmid  7015  plyval  15204  bdunexb  15860  bj-unexg  15861