Theorem uneq12d 3360

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 3354	. 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 1395   ∪ cun 3196

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-un 3202

This theorem is referenced by:  disjpr2  3731  diftpsn3  3812  iunxprg  4049  undifexmid  4281  exmidundif  4294  exmidundifim  4295  exmid1stab  4296  suceq  4497  rnpropg  5214  fntpg  5383  foun  5599  fnimapr  5702  fprg  5832  fsnunfv  5850  fsnunres  5851  tfrlemi1  6493  tfr1onlemaccex  6509  tfrcllemaccex  6522  ereq1  6704  undifdc  7111  unfiin  7113  djueq12  7232  fztp  10306  fzsuc2  10307  fseq1p1m1  10322  ennnfonelemg  13017  ennnfonelemp1  13020  ennnfonelem1  13021  ennnfonelemnn0  13036  setsvalg  13105  setsfun0  13111  setsresg  13113  setsslid  13126  prdsex  13345  prdsval  13349  psrval  14673  lgsquadlem2  15800  vtxdfifiun  16108