Theorem uneq12i 3301

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 3298	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)

Syntax hints:   = wceq 1363   ∪ cun 3141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-v 2753  df-un 3147

This theorem is referenced by:  indir  3398  difundir  3402  symdif1  3414  unrab  3420  rabun2  3428  dfif6  3550  dfif3  3561  unopab  4096  xpundi  4696  xpundir  4697  xpun  4701  dmun  4848  resundi  4934  resundir  4935  cnvun  5048  rnun  5051  imaundi  5055  imaundir  5056  dmtpop  5118  coundi  5144  coundir  5145  unidmrn  5175  dfdm2  5177  mptun  5361  fpr  5713  fvsnun2  5729  sbthlemi5  6977  djuunr  7082  djuun  7083  casedm  7102  djudm  7121  djuassen  7233  fz0to3un2pr  10140  fz0to4untppr  10141  fzo0to42pr  10237