Theorem uneq1 2177

Description: Equality theorem for union of two classes.

Assertion

Ref	Expression
uneq1	|- (A = B -> (A u. C) = (B u. C))

Proof of Theorem uneq1

Step	Hyp	Ref	Expression
1		eleq2 1535	. . . 4 |- (A = B -> (x e. A <-> x e. B))
2	1	orbi1d 615	. . 3 |- (A = B -> ((x e. A \/ x e. C) <-> (x e. B \/ x e. C)))
3		elun 2173	. . 3 |- (x e. (A u. C) <-> (x e. A \/ x e. C))
4		elun 2173	. . 3 |- (x e. (B u. C) <-> (x e. B \/ x e. C))
5	2, 3, 4	3bitr4g 555	. 2 |- (A = B -> (x e. (A u. C) <-> x e. (B u. C)))
6	5	eqrdv 1473	1 |- (A = B -> (A u. C) = (B u. C))

Syntax hints:   -> wi 3   \/ wo 222   = wceq 956   e. wcel 958   u. cun 2045

This theorem is referenced by:  uneq2 2178  uneq12 2179  uneq1i 2180  uneq1d 2183  unineq 2255  prprc1 2452  uniprg 2516  unexb 2873  suceq 3034  pwfilemOLD 4570  unxpdom 4844  sshjvalt 9320  spanunt 9468

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050

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