Theorem uneq12d 3319

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 3313	. 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 3155

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161

This theorem is referenced by:  disjpr2  3687  diftpsn3  3764  iunxprg  3998  undifexmid  4227  exmidundif  4240  exmidundifim  4241  exmid1stab  4242  suceq  4438  rnpropg  5150  fntpg  5315  foun  5526  fnimapr  5624  fprg  5748  fsnunfv  5766  fsnunres  5767  tfrlemi1  6399  tfr1onlemaccex  6415  tfrcllemaccex  6428  ereq1  6608  undifdc  6994  unfiin  6996  djueq12  7114  fztp  10170  fzsuc2  10171  fseq1p1m1  10186  ennnfonelemg  12645  ennnfonelemp1  12648  ennnfonelem1  12649  ennnfonelemnn0  12664  setsvalg  12733  setsfun0  12739  setsresg  12741  setsslid  12754  prdsex  12971  prdsval  12975  psrval  14296  lgsquadlem2  15403