Theorem uneq12d 3362

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 3356	. 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 1397   ∪ cun 3198

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204

This theorem is referenced by:  disjpr2  3733  diftpsn3  3814  iunxprg  4051  undifexmid  4283  exmidundif  4296  exmidundifim  4297  exmid1stab  4298  suceq  4499  rnpropg  5216  fntpg  5386  foun  5602  fnimapr  5706  fprg  5837  fsnunfv  5855  fsnunres  5856  tfrlemi1  6498  tfr1onlemaccex  6514  tfrcllemaccex  6527  ereq1  6709  undifdc  7116  unfiin  7118  djueq12  7238  fztp  10313  fzsuc2  10314  fseq1p1m1  10329  ennnfonelemg  13029  ennnfonelemp1  13032  ennnfonelem1  13033  ennnfonelemnn0  13048  setsvalg  13117  setsfun0  13123  setsresg  13125  setsslid  13138  prdsex  13357  prdsval  13361  psrval  14686  lgsquadlem2  15813  vtxdfifiun  16154  trlsegvdegfi  16324