Theorem uneq12d 1612

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 1610	. 2 |- (ph -> (A u. C) = (B u. C))
3		uneq12d.2	. . 3 |- (ph -> C = D)
4	3	uneq2d 1611	. 2 |- (ph -> (B u. C) = (B u. D))
5	2, 4	eqtrd 1128	1 |- (ph -> (A u. C) = (B u. D))

Syntax hints:   -> wi 2   = wceq 1091   u. cun 1485

This theorem is referenced by:  ereq 3206  mapunen 3397

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490

metamath.org