Theorem uneq12d 3359

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 3353	. 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 3195

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 2801  df-un 3201

This theorem is referenced by:  disjpr2  3730  diftpsn3  3809  iunxprg  4046  undifexmid  4278  exmidundif  4291  exmidundifim  4292  exmid1stab  4293  suceq  4494  rnpropg  5211  fntpg  5380  foun  5596  fnimapr  5699  fprg  5829  fsnunfv  5847  fsnunres  5848  tfrlemi1  6489  tfr1onlemaccex  6505  tfrcllemaccex  6518  ereq1  6700  undifdc  7102  unfiin  7104  djueq12  7222  fztp  10291  fzsuc2  10292  fseq1p1m1  10307  ennnfonelemg  12995  ennnfonelemp1  12998  ennnfonelem1  12999  ennnfonelemnn0  13014  setsvalg  13083  setsfun0  13089  setsresg  13091  setsslid  13104  prdsex  13323  prdsval  13327  psrval  14651  lgsquadlem2  15778  vtxdfifiun  16083