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Theorem mpteq12 5234
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 1910 . 2 (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶)
2 mpteq12f 5230 . 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 395  wal 1538   = wceq 1540  wral 3061  cmpt 5225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-opab 5206  df-mpt 5226
This theorem is referenced by:  mpteq1OLD  5236  mpteqb  7035  fmptcof  7150  mapxpen  9183  prodeq2w  15946  prdsdsval2  17529  prdsdsval3  17530  ablfac2  20109  mdetunilem9  22626  mdetmul  22629  xkocnv  23822  voliun  25589  itgeq1fOLD  25807  itgeq2  25813  iblcnlem  25824  bddiblnc  25877  esumeq2  34037  esumcvg  34087  dvtan  37677
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