Theorem uneq12d 3314

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 3308	. 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 1364   ∪ cun 3151

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157

This theorem is referenced by:  disjpr2  3682  diftpsn3  3759  iunxprg  3993  undifexmid  4222  exmidundif  4235  exmidundifim  4236  exmid1stab  4237  suceq  4433  rnpropg  5145  fntpg  5310  foun  5519  fnimapr  5617  fprg  5741  fsnunfv  5759  fsnunres  5760  tfrlemi1  6385  tfr1onlemaccex  6401  tfrcllemaccex  6414  ereq1  6594  undifdc  6980  unfiin  6982  djueq12  7098  fztp  10144  fzsuc2  10145  fseq1p1m1  10160  ennnfonelemg  12560  ennnfonelemp1  12563  ennnfonelem1  12564  ennnfonelemnn0  12579  setsvalg  12648  setsfun0  12654  setsresg  12656  setsslid  12669  prdsex  12880  psrval  14152