Theorem uneq12d 3364

Description: Equality deduction for union of two classes. (Contributed by NM, 29-Sep-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Hypotheses

Ref	Expression
uneq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
uneq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
uneq12d	⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Proof of Theorem uneq12d

Step	Hyp	Ref	Expression
1		uneq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		uneq12d.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐷)
3		uneq12 3358	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   = wceq 1398   ∪ cun 3199

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205

This theorem is referenced by:  disjpr2  3737  diftpsn3  3819  iunxprg  4056  undifexmid  4289  exmidundif  4302  exmidundifim  4303  exmid1stab  4304  suceq  4505  rnpropg  5223  fntpg  5393  foun  5611  fnimapr  5715  fprg  5845  fsnunfv  5863  fsnunres  5864  tfrlemi1  6541  tfr1onlemaccex  6557  tfrcllemaccex  6570  ereq1  6752  undifdc  7159  unfiin  7161  djueq12  7281  fztp  10356  fzsuc2  10357  fseq1p1m1  10372  ennnfonelemg  13085  ennnfonelemp1  13088  ennnfonelem1  13089  ennnfonelemnn0  13104  setsvalg  13173  setsfun0  13179  setsresg  13181  setsslid  13194  prdsex  13413  prdsval  13417  psrval  14742  lgsquadlem2  15877  vtxdfifiun  16218  trlsegvdegfi  16388