Theorem uneq12d 3378

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 3372	. 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 1398   ∪ cun 3212

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-un 3218

This theorem is referenced by:  disjpr2  3758  diftpsn3  3840  iunxprg  4077  undifexmid  4311  exmidundif  4324  exmidundifim  4325  exmid1stab  4326  suceq  4528  rnpropg  5247  fntpg  5417  fresaunres2disj  5550  foun  5638  fnimapr  5742  fprg  5872  fsnunfv  5890  fsnunres  5891  tfrlemi1  6576  tfr1onlemaccex  6592  tfrcllemaccex  6605  ereq1  6787  mapunen  7117  undifdc  7197  unfiin  7199  djueq12  7343  fztp  10434  fzsuc2  10435  fseq1p1m1  10450  ennnfonelemg  13238  ennnfonelemp1  13241  ennnfonelem1  13242  ennnfonelemnn0  13257  setsvalg  13326  setsfun0  13332  setsresg  13334  setsslid  13347  prdsex  13566  prdsval  13570  psrval  14926  lgsquadlem2  16063  vtxdfifiun  16404  trlsegvdegfi  16574