Theorem uneq12d 3318

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 3312	. 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  3686  diftpsn3  3763  iunxprg  3997  undifexmid  4226  exmidundif  4239  exmidundifim  4240  exmid1stab  4241  suceq  4437  rnpropg  5149  fntpg  5314  foun  5523  fnimapr  5621  fprg  5745  fsnunfv  5763  fsnunres  5764  tfrlemi1  6390  tfr1onlemaccex  6406  tfrcllemaccex  6419  ereq1  6599  undifdc  6985  unfiin  6987  djueq12  7105  fztp  10153  fzsuc2  10154  fseq1p1m1  10169  ennnfonelemg  12620  ennnfonelemp1  12623  ennnfonelem1  12624  ennnfonelemnn0  12639  setsvalg  12708  setsfun0  12714  setsresg  12716  setsslid  12729  prdsex  12940  psrval  14220  lgsquadlem2  15319