Theorem uneq12i 3288

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 3285	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
4	1, 2, 3	mp2an 426	1 ⊢ (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷)

Syntax hints:   = wceq 1353   ∪ cun 3128

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134

This theorem is referenced by:  indir  3385  difundir  3389  symdif1  3401  unrab  3407  rabun2  3415  dfif6  3537  dfif3  3548  unopab  4083  xpundi  4683  xpundir  4684  xpun  4688  dmun  4835  resundi  4921  resundir  4922  cnvun  5035  rnun  5038  imaundi  5042  imaundir  5043  dmtpop  5105  coundi  5131  coundir  5132  unidmrn  5162  dfdm2  5164  mptun  5348  fpr  5699  fvsnun2  5715  sbthlemi5  6960  djuunr  7065  djuun  7066  casedm  7085  djudm  7104  djuassen  7216  fz0to3un2pr  10123  fz0to4untppr  10124  fzo0to42pr  10220