Theorem uneq12d 3332

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 3326	. 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 1373   ∪ cun 3168

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174

This theorem is referenced by:  disjpr2  3701  diftpsn3  3779  iunxprg  4013  undifexmid  4244  exmidundif  4257  exmidundifim  4258  exmid1stab  4259  suceq  4456  rnpropg  5170  fntpg  5338  foun  5552  fnimapr  5651  fprg  5779  fsnunfv  5797  fsnunres  5798  tfrlemi1  6430  tfr1onlemaccex  6446  tfrcllemaccex  6459  ereq1  6639  undifdc  7035  unfiin  7037  djueq12  7155  fztp  10215  fzsuc2  10216  fseq1p1m1  10231  ennnfonelemg  12844  ennnfonelemp1  12847  ennnfonelem1  12848  ennnfonelemnn0  12863  setsvalg  12932  setsfun0  12938  setsresg  12940  setsslid  12953  prdsex  13171  prdsval  13175  psrval  14498  lgsquadlem2  15625