Theorem uneq12d 3288

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 3282	. 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 1353   ∪ cun 3125

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131

This theorem is referenced by:  disjpr2  3653  diftpsn3  3730  iunxprg  3962  undifexmid  4188  exmidundif  4201  exmidundifim  4202  suceq  4396  rnpropg  5100  fntpg  5264  foun  5472  fnimapr  5568  fprg  5691  fsnunfv  5709  fsnunres  5710  tfrlemi1  6323  tfr1onlemaccex  6339  tfrcllemaccex  6352  ereq1  6532  undifdc  6913  unfiin  6915  djueq12  7028  fztp  10048  fzsuc2  10049  fseq1p1m1  10064  ennnfonelemg  12371  ennnfonelemp1  12374  ennnfonelem1  12375  ennnfonelemnn0  12390  setsvalg  12459  setsfun0  12465  setsresg  12467  setsslid  12479  exmid1stab  14310