Theorem uneq12d 2188

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 2186	. 2 |- (ph -> (A u. C) = (B u. C))
3		uneq12d.2	. . 3 |- (ph -> C = D)
4	3	uneq2d 2187	. 2 |- (ph -> (B u. C) = (B u. D))
5	2, 4	eqtrd 1510	1 |- (ph -> (A u. C) = (B u. D))

Syntax hints:   -> wi 3   = wceq 958   u. cun 2048

This theorem is referenced by:  oarec 4202  oaabs 4258  ereq 4273  mapunen 4508  icoun 6414  snunioo 6416  sumeq1 6982  sumeq2 6985  dffsum 6998  dfisum 7191  alephadd 7584  ispgrag 10750

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-un 2053

Copyright terms: Public domain