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Theorem mpteq12 5240
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 5236 . 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 5231
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 5211  df-mpt 5232
This theorem is referenced by:  mpteq1OLD  5242  mpteqb  7017  fmptcof  7130  mapxpen  9145  prodeq2w  15860  prdsdsval2  17434  prdsdsval3  17435  ablfac2  20000  mdetunilem9  22342  mdetmul  22345  xkocnv  23538  voliun  25295  itgeq1f  25513  itgeq2  25519  iblcnlem  25530  bddiblnc  25583  esumeq2  33320  esumcvg  33370  dvtan  36841
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