Theorem uneq12 2175

Description: Equality theorem for union of two classes.

Assertion

Ref	Expression
uneq12	|- ((A = B /\ C = D) -> (A u. C) = (B u. D))

Proof of Theorem uneq12

Step	Hyp	Ref	Expression
1		uneq1 2173	. 2 |- (A = B -> (A u. C) = (B u. C))
2		uneq2 2174	. 2 |- (C = D -> (B u. C) = (B u. D))
3	1, 2	sylan9eq 1524	1 |- ((A = B /\ C = D) -> (A u. C) = (B u. D))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   u. cun 2041

This theorem is referenced by:  uneq12i 2178  un00 2302  opthprc 3216  unixp 3509  fnun 3586  oarec 4186  pm54.43 4552  trcl 4625

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046

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