Theorem uneq12d 2185

Description: Equality deduction for intersection of two classes.

Hypotheses

Ref	Expression
uneq1d.1	|- (ph -> A = B)
uneq12d.2	|- (ph -> C = D)

Assertion

Ref	Expression
uneq12d	|- (ph -> (A u. C) = (B u. D))

Proof of Theorem uneq12d

Step	Hyp	Ref	Expression
1		uneq1d.1	. . 3 |- (ph -> A = B)
2	1	uneq1d 2183	. 2 |- (ph -> (A u. C) = (B u. C))
3		uneq12d.2	. . 3 |- (ph -> C = D)
4	3	uneq2d 2184	. 2 |- (ph -> (B u. C) = (B u. D))
5	2, 4	eqtrd 1507	1 |- (ph -> (A u. C) = (B u. D))

Syntax hints:   -> wi 3   = wceq 956   u. cun 2045

This theorem is referenced by:  oarec 4196  oaabs 4252  ereq 4267  mapunen 4502  icoun 6413  snunioo 6415  sumeq1 6982  sumeq2 6985  dffsum 6998  dfisum 7191  alephadd 7582  ispgrag 10779

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

Copyright terms: Public domain