Theorem uneq12i 3287

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 3284	. 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 3127

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 2739  df-un 3133

This theorem is referenced by:  indir  3384  difundir  3388  symdif1  3400  unrab  3406  rabun2  3414  dfif6  3536  dfif3  3547  unopab  4082  xpundi  4682  xpundir  4683  xpun  4687  dmun  4834  resundi  4920  resundir  4921  cnvun  5034  rnun  5037  imaundi  5041  imaundir  5042  dmtpop  5104  coundi  5130  coundir  5131  unidmrn  5161  dfdm2  5163  mptun  5347  fpr  5698  fvsnun2  5714  sbthlemi5  6959  djuunr  7064  djuun  7065  casedm  7084  djudm  7103  djuassen  7215  fz0to3un2pr  10122  fz0to4untppr  10123  fzo0to42pr  10219