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Theorem mpteq12 5239
Description: An equality theorem for the maps-to notation. (Contributed by NM, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐷(𝑥)

Proof of Theorem mpteq12
StepHypRef Expression
1 ax-5 1913 . 2 (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶)
2 mpteq12f 5235 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
31, 2sylan 580 1 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wral 3061  cmpt 5230
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-opab 5210  df-mpt 5231
This theorem is referenced by:  mpteq1OLD  5241  mpteqb  7014  fmptcof  7124  mapxpen  9139  prodeq2w  15852  prdsdsval2  17426  prdsdsval3  17427  ablfac2  19953  mdetunilem9  22113  mdetmul  22116  xkocnv  23309  voliun  25062  itgeq1f  25280  itgeq2  25286  iblcnlem  25297  bddiblnc  25350  esumeq2  33022  esumcvg  33072  dvtan  36526
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