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Theorem uneq12d 3125
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 (𝜑𝐴 = 𝐵)
uneq12d.2 (𝜑𝐶 = 𝐷)
Assertion
Ref Expression
uneq12d (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Proof of Theorem uneq12d
StepHypRef Expression
1 uneq1d.1 . 2 (𝜑𝐴 = 𝐵)
2 uneq12d.2 . 2 (𝜑𝐶 = 𝐷)
3 uneq12 3119 . 2 ((𝐴 = 𝐵𝐶 = 𝐷) → (𝐴𝐶) = (𝐵𝐷))
41, 2, 3syl2anc 397 1 (𝜑 → (𝐴𝐶) = (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  cun 2942
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949
This theorem is referenced by:  disjpr2  3461  diftpsn3  3532  suceq  4166  rnpropg  4827  fntpg  4982  foun  5172  fnimapr  5260  fprg  5373  fsnunfv  5390  fsnunres  5391  tfrlemi1  5976  ereq1  6143  fztp  9041  fzsuc2  9042  fseq1p1m1  9057
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