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Theorem mpteq12 4664
 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 1827 . 2 (𝐴 = 𝐶 → ∀𝑥 𝐴 = 𝐶)
2 mpteq12f 4661 . 2 ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
31, 2sylan 487 1 ((𝐴 = 𝐶 ∧ ∀𝑥𝐴 𝐵 = 𝐷) → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475  ∀wral 2896   ↦ cmpt 4643 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ral 2901  df-opab 4644  df-mpt 4645 This theorem is referenced by:  mpteq1  4665  mpteqb  6207  fmptcof  6304  mapxpen  8011  prodeq2w  14481  prdsdsval2  15967  prdsdsval3  15968  ablfac2  18311  mdetunilem9  20245  mdetmul  20248  xkocnv  21427  voliun  23129  itgeq1f  23344  itgeq2  23350  iblcnlem  23361  esumeq2  29425  esumcvg  29475  dvtan  32630  bddiblnc  32650
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