Theorem uneq12i 3273

Description: Equality inference for union of two classes. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
uneq1i.1	⊢ 𝐴 = 𝐵
uneq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
uneq12i	⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)

Proof of Theorem uneq12i

Step	Hyp	Ref	Expression
1		uneq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		uneq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		uneq12 3270	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
4	1, 2, 3	mp2an 423	1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)

Syntax hints:   = wceq 1343   ∪ cun 3113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119

This theorem is referenced by:  indir  3370  difundir  3374  symdif1  3386  unrab  3392  rabun2  3400  dfif6  3521  dfif3  3532  unopab  4060  xpundi  4659  xpundir  4660  xpun  4664  dmun  4810  resundi  4896  resundir  4897  cnvun  5008  rnun  5011  imaundi  5015  imaundir  5016  dmtpop  5078  coundi  5104  coundir  5105  unidmrn  5135  dfdm2  5137  mptun  5318  fpr  5666  fvsnun2  5682  sbthlemi5  6922  djuunr  7027  djuun  7028  casedm  7047  djudm  7066  djuassen  7169  fz0to3un2pr  10054  fz0to4untppr  10055  fzo0to42pr  10151