Theorem uneq12d 3359

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 3353	. 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 1395   ∪ cun 3195

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201

This theorem is referenced by:  disjpr2  3730  diftpsn3  3808  iunxprg  4045  undifexmid  4276  exmidundif  4289  exmidundifim  4290  exmid1stab  4291  suceq  4490  rnpropg  5204  fntpg  5373  foun  5587  fnimapr  5687  fprg  5815  fsnunfv  5833  fsnunres  5834  tfrlemi1  6468  tfr1onlemaccex  6484  tfrcllemaccex  6497  ereq1  6677  undifdc  7074  unfiin  7076  djueq12  7194  fztp  10262  fzsuc2  10263  fseq1p1m1  10278  ennnfonelemg  12960  ennnfonelemp1  12963  ennnfonelem1  12964  ennnfonelemnn0  12979  setsvalg  13048  setsfun0  13054  setsresg  13056  setsslid  13069  prdsex  13288  prdsval  13292  psrval  14615  lgsquadlem2  15742