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  4277  exmidundif  4290  exmidundifim  4291  exmid1stab  4292  suceq  4493  rnpropg  5208  fntpg  5377  foun  5593  fnimapr  5696  fprg  5826  fsnunfv  5844  fsnunres  5845  tfrlemi1  6484  tfr1onlemaccex  6500  tfrcllemaccex  6513  ereq1  6695  undifdc  7094  unfiin  7096  djueq12  7214  fztp  10282  fzsuc2  10283  fseq1p1m1  10298  ennnfonelemg  12982  ennnfonelemp1  12985  ennnfonelem1  12986  ennnfonelemnn0  13001  setsvalg  13070  setsfun0  13076  setsresg  13078  setsslid  13091  prdsex  13310  prdsval  13314  psrval  14638  lgsquadlem2  15765