Theorem uneq12d 3384

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 3378	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))
4	1, 2, 3	syl2anc 415	1 ⊢ (𝜑 → (𝐴 ∪ 𝐶) = (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   = wceq 1402   ∪ cun 3218

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224

This theorem is referenced by:  disjpr2  3772  diftpsn3  3854  iunxprg  4091  undifexmid  4328  exmidundif  4341  exmidundifim  4342  exmid1stab  4343  suceq  4545  rnpropg  5265  fntpg  5435  fresaunres2disj  5568  foun  5656  fnimapr  5760  fprg  5892  fsnunfv  5910  fsnunres  5911  tfrlemi1  6597  tfr1onlemaccex  6613  tfrcllemaccex  6626  ereq1  6808  mapunen  7145  undifdc  7225  unfiin  7227  djueq12  7373  fztp  10468  fzsuc2  10469  fseq1p1m1  10484  ennnfonelemg  13277  ennnfonelemp1  13280  ennnfonelem1  13281  ennnfonelemnn0  13296  setsvalg  13365  setsfun0  13371  setsresg  13373  setsslid  13386  prdsex  14155  prdsval  14156  psrval  15033  lgsquadlem2  16180  vtxdfifiun  16521  trlsegvdegfi  16691